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NEW BOUNDS ON NEARLY PERFECT MATCHINGS IN HYPERGRAPHS: HIGHER CODEGREES DO HELP

Van. H. Vu Microsoft Research Redmond, WA 98052, USA

Abstract.

Let H be a (k + 1)-uniform, D-regular hypergraph on n vertices and U (H) be the minimum number of vertices left uncovered by a matching in H. Cj (H), the j-codegree of H, is the maximum number of edges sharing a set of j vertices in common. We prove a general upper bound on U (H), based on the codegree sequence C2 (H), C3 (H) . . . . Our bound improves and generalizes many results on the topic, including those of Grable [Gra], AlonKim-Spencer [AKS], and Kostochka-R¨ odl [KR]. It also leads to a substantial improvement in several applications. The key ingredient of the proof is the so-called polynomial technique, which is a new and useful tool to prove concentration results for functions with large Lipschitz coefficient. This technique is of independent interest.

§1 INTRODUCTION §1.1 The problem and earlier results A hypergraph H(V, E) consists of a set V of vertices and a family E of subsets of V , called edges. We say that H is r-uniform if every edge has exactly r vertices. The degree of a vertex v is the number of edges containing v and H is D-regular if the degree of every vertex is D. In this paper r is fixed, and D → ∞. The asymptotic notation such as o, O, Θ etc is understood with this assumption. The codegree (codeg(u, v)) of a pair u, v of vertices is the number of edges containing both u and v. More generally, given s vertices v1 , . . . , vs , the codegree codeg(v1 , . . . , vs ) is the number of edges containing all of them. The s-codegree of H is the maximum codegree of a set of s vertices. When s = 2, we follow the literature and call the 2-codegree of H its codegree. A matching of H is a set of disjoint edges. Determining when H contains a large matching is a fundamental problem in combinatorics, with a wide range of applications. Pippenger [Pip], based on the pioneering work of R¨odl [R¨od], observed that the existence of a large matching in a regular hypergraph is guaranteed by the following simple and general assumption on the codegree. Let U(H) denote the minimum number of vertices in H left uncovered by a matching. Part of this work was done while the author was with the Institute for Advanced Study ( Priceton, NJ) and was supported by a grant from NEC and the state of New Jersey 1

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Theorem 1.1.1. ( [Pip]) Assume that H is an r-uniform and D-regular hypergraph on n vertices and codeg(H) = C = o(D), then there is a matching which covers all but o(n) vertices, i.e., U (H) = o(n). Roughly speaking, if H is regular with small degree relative to to its degree, then H has a near perfect matching. In [Spe], Spencer reproved this theorem by a different method. Pippenger’s theorem, however, does not supply an explicit estimate for the error term o(n). For instance, it is not clear how C and D contribute to U(H). Sharpening this error term, which is useful for many applications, is a challenging problem which attracted the attention of several researchers ([Gra], [Gra2], [AKS], [KR], [Kuz], [Kuz2]). In particular, Grable [Gra] proved the following theorem. Theorem 1.1.2. If H is a (k + 1)-uniform, hypergraph ¢ on n vertices with ¡ D-regular n 1/(2k+1+o(1)) . ) codeg(H) = C = o(D/ log n), then U(H) = O n( C log D Alon, Kim and Spencer [AKS] proved the following theorem which improves Grable’s bound for the special case when codeg(H) = C = 1, i.e., the hypergraph is linear. Theorem 1.1.3. If H is a (k + 1)-uniform, D-regular hypergraph on n vertices with ¡ 1 1/k ¢ ¡ 1 1/2 ¢ codeg(H) = C = 1, then U(H) = O n( D ) for k ≥ 3 and U(H) = O n( D ) log3/2 D for k = 2. In [KR], Kostochka and R¨odl extended Theorem 1.1.3 to hypergraphs with moderate codegrees. Using Theorem 1.1.3 as a key lemma, they proved that Theorem 1.1.4. Let δ and γ be two arbitrary positive numbers. If H is a (k + 1)-uniform (k¡ ≥ 3), D-regular hypergraph on n vertices with codegree C ≤ D1−γ , then U (H) = ¢ C 1/k−δ O n( D ) . In view of Theorem 1.1.3, it is natural to think¡ that the¢ constants δ and γ in Theorem C 1/k 1.1.4 should be removed to give a better bound O ( D ) n . This bound would be a direct generalization of Theorem 1.1.3. The reader would have already observed that in all quoted results the codegree (which is, by our definition, the 2-codegree) seems to be the only parameter which matters. This makes us wonder whether the higher codegrees play some role in bounding U (H). Our paper is motivated by the following question: Can one use information about higher codegrees to improve the bound on U (H) ? The general answer we obtain to this question is affirmative. This is interesting in both theoretical and practical points of view. First, the key fact that higher codegrees do help may be true in several other problems (coloring, for instance) and may lead to a new direction for further research in the field. Second, in many natural applications, the higher codegrees are easy to compute, and our results show that they can be used to provide a substantial improvement on earlier results. The first goal of our paper is to give a way to quantify the contribution of the higher codegrees in the bound on U(H). This is done in Theorem 1.2.2. A special case of this theorem (Theorem 1.2.1) enables us to remove the unwanted constants γ and δ in Kostochka-R¨odl’s result, with the cost of a logarithmic term. We shall discuss several applications to illustrate the fact that information about higher codegrees does indeed change the picture. Our second goal is to present, via the proofs, a new technique which seems to be very helpful in proving large deviation bounds. Consider a product space generalized by independent random variables x1 , . . . , xm , and a function Y = Y (x1 , . . . , xm ). When each

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variable has only a small effect on Y , i.e., changing xi changes Y by a small amount, one can prove that Y is strongly concentrated around its mean, using Talagrand type inequalities or various martingale arguments. However, once there are many variables with rather large effects, these methods frequently fail (as they do in our setting). Our new method, what we call the polynomial method, is a tool developed in order to overcome this difficulty in certain situations. This method is based on a new type of concentration results developed in [KV] and[Vu1] and is of independent interest. The rest of the paper is organized as follows. In the next subsection (§1.2), we present our new results and the applications. In §1.3, we describe our key tool, the polynomial method, and the main ideas of the proofs. Section 2 provides the main tools, which include a Talagrand’s type inequality and a new concentration result on polynomials. We shall make a comparison between the two concentration results from the applicability point of view, and point out why our new concentration result is crucial in the proofs. The proof of Theorem 1.2.1 is presented in Sections 3 and 4. In Section 5, we extend this proof to obtain the more general result, Theorem 1.2.2. §1.2 New results and applications ˜ In this section, we use the notation O(.), which is frequently used in theoretical computer ˜ ), if there is a constant c so that science. We say that a quantity X is of order O(f X ≤ f logc D. To this end, let D denote the degree of the hypergraph H, from which we would like to extract a large matching. In the rest of the paper, we assume that D is sufficiently large, whenever needed. Theorem 1.2.1. For all k ≥ 3 the following holds. If H ¡is a (k +¢ 1)-uniform, D-regular ˜ n( C )1/k . hypergraph on n vertices with codegree C, then U (H) = O D Theorem 1.2.1 improves upon Kostochka and R¨odl’s result in two ways. First, the constant δ (a polynomial term) is removed at the cost of a logarithmic term. Second, the condition C ≤ D1−γ is no longer needed. Up to the negligible polylogarithmic term, Theorem 1.2.1 could also be seen as a natural generalization of Alon-Kim-Spencer’s result (Theorem 1.1.3). Theorem 1.2.1 is a special case of the following general result, which is the first goal of this paper. To begin, let us recall that Cj (H) denotes the j-codegree of H; in particular C2 (H) is the codegree of H. Theorem 1.2.2. Let H be a (k + 1)-uniform, D-regular hypergraph on n vertices. Assume that for some s ≤ k + 1 there are D1 = D ≥ D2 ≥ · · · ≥ Ds > 0 and x > 0 such that (1) For every j < s, Cj (H) ≤ Dj . (2) For every j ≥ s, Cj (H) ≤ Ds . (3) x3 ≤ Dj /Dj+1 for all j ≤ s − 1. (4) xk−s+2 ≤ Ds−1 /Ds . ˜ −1 n). Then U (H) = O(x Remark. Both Theorems 1.2.1 and 1.2.2 still hold in the case when the hypergraph is not regular, but is close to be one (see the Bite Lemma and Lemma 5.1). Instead of D-regularity, ¡ ¢ ˜ ( C )1/k it suffices to assume that every degree of H is between D(1 ± ²0 ), where ²0 = Θ D ˜ −1 ) for Theorem 1.2.2 (the precise statements can be seen for Theorem 1.2.1 and ²0 = Θ(x in §3 and §5, resp). It was shown that Theorems 1.1.3 and 1.1.4 hold under a similar, but ˜ 1 )1/2 ). So among all results, ours allows the stronger assumption, where ²0 need to be Θ(( D largest fluctuation on the degrees.

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To see that Theorem 1.2.2 implies Theorem 1.2.1 as a special case, consider a hypergraph 1 1/k H as in Theorem 1.2.1 and set s = 2, D1 = D, D2 = Ds = C and x = ( D . It is D2 ) obvious that the assumptions (1), (2) and (4) of Theorem 1.2.2 are satisfied. Furthermore, assumption (3) also holds since k ≥ 3. Theorem 1.2.2 then implies that there is a matching ˜ C )1/k n) vertices of H, which is the conclusion of ˜ D1 )−1/k n) = O(( that covers all but O(( D2 D Theorem 1.2.1. One may wonder if the exponent 1/k in Theorem 1.2.1 can be improved to yield a better bound. In our opinion, this would be quite difficult, given the current method, and we shall address this issue in the next section. As already mentioned, in many applications the hypergraph is very special and thus its codegrees (of any order) are easy to compute. In these cases, a proper use of Theorem 1.2.2 frequently gives a bound much better than what one has using Theorem 1.2.1. In the following, an example and two applications are given to illustrate this fact. Example. Assume that k is large (k ≥ 10, say) and H is a (k + 1)-uniform, Dregular hypergraph with C = C2 (H) = D1−β for some positive constant β < 3/(k + 2). If we have no other information, then the best we can achieve using Theorem 1.2.1 is that 1/k ˜ ˜ −β/k n). On the other hand, if we also know that the 3U(H) = O((C/D) n) = O(D codegree of H is small, say O(1), then this bound can be improved significantly. Indeed, set s = 3, D1 = D, D2 = C, D3 = O(1), and x = Dβ/3 . We can now apply Theorem 1.2.2. applies to get that 1/3 ˜ −1 n) = O(D ˜ −β/3 n) = O((C/D) ˜ U(H) = O(x n), −β/k 1/k ˜ ˜ which is clearly much better than the previous bound O(D n) = O((C/D) n). The most remarkable fact about this bound is that the crucial exponent 1/3 does not depend on k.

Partial Steiner systems A partial Steiner system S(t, r, m) is an r-uniform hypergraph on m vertices so that every ¡set¢ of¡ t¢ vertices is contained in at most one edge. Any partial Steiner system has at r most m od], confirming a famous conjecture of Erd˝os t / t edges. In his original paper [R¨ and Hanani, R¨odl proved that Theorem 1.2.3. Assume that t and¡ r ¢are ¡r¢fixed and m → ∞, then there is a S(t, r, m) partial Steiner system with (1 − o(1)) m t / t edges. The proof of R¨odl formalizes the use of the semi-random method (which is also called the R¨odl nibble method) and had a great influence on extremal combinatorics in the last decade (see [Kah] for a survey). ¡ ¢ t Define a special hypergraph H in the following way. H has n = m t = Θ(m ) vertices, where each vertex represents a t-tuple. An edge of H consists of those t-tuples which are ¡¢ ¡ ¢ subsets of the same r set. Thus H is rt -uniform and m−t -regular. Moreover, the codegree r−t ¡m−t−1¢ of H is r−t−1 = C. The existence of a large partial Steiner system then corresponds to the existence of a large matching in H. In this way, Theorem 1.1.1 [Pip] can be seen as a generalization of Theorem 1.2.3. Moreover, all the results mentioned in §1.1 have a corollary which sharpens Theorem 1.2.3. For instance, Theorem 1.1.4 [KR] implies that ¡ ¢ r there is a partial Steiner system which covers all but O mt−1/(( t )−1)+δ t-tuples [KR]. Theorem 1.2.1 implies the following corollary, which improves Kostochka-R¨odl’s bound by removing δ.

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Corollary 1.2.4. Assume that t and r are fixed and m → ∞. Then there is a Steiner ¡ ¢ r ˜ mt−1/(( t )−1) t-tuples. Moreover, this system system on m points which covers all but O can be constructed by a polynomial randomized algorithm which succeeds with probability 1 − o(1). The algorithmic part of the corollary is due to the remark at the end of §4.3 (here polynomial means polynomial in m). In Kostochka-R¨odl’s paper, it is not clear that their bound can be obtained by an efficient algorithm. Now assume that r ≥ t + 3, we show that Theorem 1.2.2 gives a substantially better bound. Corollary 1.2.5. Assume that t and r are fixed and m → ∞. Then there is a Steiner sys¡ ¢ r ˜ mt−(r−t)/3(( t )−1) t-tuples. Moreover, this system tem on m points which covers all but O can be constructed by a polynomial randomized algorithm which succeeds with probability 1 − o(1). Proof. We prove the existence part only (the algorithmic part follows from ¡j ¢ the remark ¡ ¢ at the end of Section 5). In the hypergraph H defined on t-tuples, for any t ≥ l > j−1 t , Cl (H) = Θ(mr−j ). ¡¢ r Set x = cm(r−t)/3(( t )−1) , s = rt , where c is a properly chosen positive constant. For ¡¢ r r any 2 ≤ l ≤ r , set D = m(( t )−l)(r−t)/(( t )−1) . As usual D = D. To complete the proof t

l

1

we need to show that the assumptions of Theorem 1.2.2 are satisfied with these parameters. The only non-trivial assumption is Dl ≥ Cl (for all l). To verify this assumption, it suffices to prove that µ ¶ µ ¶ r j r−t ¡r ¢ ( − ) ≥ r − j, t t − 1 t which is (via a routine simplification) equivalent to (j − t)([r]t − [j]t ) ≥ (r − j)([j]t − [t]t ), where [x]t = x(x − 1) . . . (x − t + 1). The proof of this is a little bit tricky. Let f (x) = [x]t , it is easy to show that f ”(x) ≥ 0 if x ≥ t which implies that f 0 (x) is monotone increasing in [t, r]. On the other hand, [r]t − [j]t = (r − j)f 0 (y) and [j]t − [t]t = (j − t)f 0 (z) for some y ∈ [j, r] and z ∈ [t, j]. Since y ≥ z ≥ t, f 0 (y) ≥ f 0 (z) and the proof is finished. ¤ To finish this section, let us consider another application of our results in the following problem, suggested by M. Krivelevich (personal communication). Edge cover in random graphs. Let G0 be a fixed, small graph and G be a much bigger graph. We want to cover the edges of G by edge disjoint copies of G0 . Let f (G0 , G) denote the minimum number of edges left uncovered. Now consider G(n, p), the random graph on n points, we want to find a quantity K(n, G0 , p) such that a.s, f (G0 , G(n, p)) ≤ K(n, G0 , p). This problem can be seen as a variant of the popular vertex cover problem (when one wants to cover the vertices by vertex disjoint copies of G0 ). For instance, if G0 is a circle of arbitrary length, then K(n, G0 , p) should be at least linear in n, since with probability close to 1, about half of the vertices of G(n, p) have odd degree, and therefore there will be at least one uncovered edge from these vertices. It is not

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clear whether this lower bound is sharp; however, this parity argument already indicates that despite the similarity in formalization, the nature of the edge cover problem and that of the vertex cover problem are very different. Using Theorem 1.2.2, we are able to give a non-trivial upper bound on K(n, G0 , p) for many G0 . For the sake of simplicity, let us consider a special case where p is a constant and G0 = Circlek+1 is a circle of length k + 1. It is easy to modify the argument for different p and G0 . Consider a graph G from G(n, p); define a hypergraph H as follows. The vertex set of H is the edge set of G; the edges of H are the circles of length k + 1 of G. Then a collection of edge disjoint copies of circles of length k + 1 is a matching in H. It is clear that H is k + 1-uniform. Moreover, it is simple to verify that a.s. the following hold ¡ ¢ • V (H) ≈ n2 p = Θ(n2 ), • For all v ∈ V (H), degv ≈ [n]k−1 pk = D = Θ(nk−1 ), where [n]r = n(n−1) . . . (n−r+1). • Cj (H) = O(nk−j pk−j+1 ) for all j = 2, 3, . . . , k. Assume that the above three properties hold. Set s = k − 1, D1 = D and Dj = ank−j , x = cn1/3 , where a and c are properly chosen constants, one can show that the assumptions of Theorem 1.2.2 are satisfied (notice that H is not a regular hypergraph, but a.s. all degrees are very close to D so Theorem 1.2.2 still applies (see the remark following Theorem 1.2.2). ˜ −1 n2 ) = O(n ˜ 5/3 ). This means that K(n, Circlek+1 , p) = O(n ˜ 5/3 ) Therefore, U(H) = O(x for any k. It is clear that using the less general Theorem 1.2.1, we could only prove a much ˜ 2−1/k ). If k = 2, then one can use Theorem weaker result that K(n, Circlek+1 , p) = O(n ˜ 3/2 ) (in this case the corresponding hypergraph is 1.1.3 to deduce a better bound of O(n linear). Corollary 1.2.6. In G(n, p) with constant p, the following holds almost surely ˜ 3/2 ) edges. (a) There is a collection of edge disjoint triangles which covers all but O(n (b) For all fixed k ≥ 4, there is a collection of edge disjoint circles of length k which ˜ 5/3 ) edges. covers all but O(n §1.3 The ideas and the polynomial method In the rest of the paper, dv and codeg(v) will denote the degree of a point v and the codegree of a set (of vertices) v, respectively. If E is a set of edges, V (E) denotes the set of vertices covered by E. It is rather tedious to prove Theorem 1.2.2 at once, so we shall first prove Theorem 1.2.1 and then extend this proof to obtain Theorem 1.2.2. Here are the main ideas of the proof of Theorem 1.2.1. We use a version of the so-called “semi-random” or “nibble” technique, initiated by R¨odl [R¨od]. In our version, we consider a “double nibble” process, which consists of two type of nibble processes: primary and secondary. In each step of any of these processes, we do the following: Take out a set of disjoint edges and add to the matching. Delete the points covered by the matching and some additional points. Continue with the remaining induced hypergraph. The set of additional points is called the error term of a step. The set of deleted points is called a nibble or a bite, according to its size. The secondary process consists of very big bites, each of which reduces the size of the C 1/k vertex set by a factor (roughly) e (≈ 2.718) and has an error term of order O(n( D ) ). k Each bite reduces the degree of the remaining hypergraph by a factor e . After having

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about log D C bites, we may expect that that the degree of the remaining hypergraph is of C 1/k ˜ ˜ order O(C) and there are O(n( ) ) vertices left. Moreover, the error accumulated in the ¡ C 1/k D D ¢ ¡ ¢ ˜ n( C )1/k . At this stage, we end the process and process is of order O n( D ) log C = O ¡ C 1/k ¢D ˜ n( ) vertices uncovered. In fact, one can adjust conclude that there are at most O D the parameters so that the major part of the set of uncovered vertices is the vertex set of the final hypergraph (see Remark 4.1.1). It is a good time to point out why the bound in Theorem 1.2.1 is hard to improve, at least by the current method. At the point where the degrees of the remaining hypergraph reduce to O(C), we should end the process since it is very hard to show that a big matching still exists. The reason is that a hypergraph with degrees of order O(C) and codegree C may not contain a big matching, and we hardly know anything else about our hypergraph except the ¢degree and codegree bounds. On the other hand, at this point, there are still ¡ C 1/k Ω (D ) n uncovered vertices. It is not clear how to achieve the desired bites at once. In our proof, we shall produce each bite as the union of several nibbles. These nibbles are provided by a primary process. Each nibble reduces the number of vertices by only a factor of (1 − θ)−1 , where θ is a small, dynamic quantity which depends on the parameters of the hypergraph at the moment. Taking about θ−1 nibbles, we would end up with a bite which reduces the vertex set by −1 a factor (1 − θ)−θ ≈ e. Tight control on the error term of each nibble will give us the desired error term for a bite. Together the picture looks as follows: There are many primary processes, each results in a big bite. The bites form the secondary process, which provides the desired matching. The secondary process was introduced for a purely technical reason in order to simplify the computation, and does not have a central role in the proof. The main difficulty of the proof involves the primary processes and is hidden in the requirement that we need to keep very tight control on each nibble. We shall have to insure that in each nibble, the size of the vertex set, the error term and the degrees of the remaining vertices behave almost exactly as expected. This, in turn, demands various strong concentration results. The proof of the hardest among these results (Claim 4.2.3. of § 4.2), requires the new method (polynomial method) mentioned in the last paragraph of Section §1.1. The polynomial method. The polynomial method can be applied as follows. Given a function X from a product space Ω generated by independent variables x1 , . . . , xm to R. In order to show that X is strongly concentrated, we first find two polynomials Y1 and Y2 of small degree such that Y1 (x) ≤ X(x) ≤ Y2 (x), for all x ∈ Ω E(Y1 ) ≈ E(X) ≈ E(Y2 ). Once Y1 and Y2 are found, all we need is to show that both of them are strongly concentrated. Since X is sandwiched between the two, X itself should also be strongly concentrated. There are two questions we need to answer: How to find the polynomials and how to prove that they are strongly concentrated ? There is no universal answer to the first question; but very often, Yi can be found by taking the first few terms in the Taylor series expansion of X, and this is exactly what we do in this paper. The answer to the second question relies on a series of new large deviation

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bounds of low degree multivariate polynomials, proven in [Vu1, Vu5] and [KV]. The fact that these bounds can give strong concentration results in many situations where standard methods do not apply is the main power of this method. We shall discuss this crucial issue in detail in the last part of the next section and the last part of § 4.2. It would be useful for the readers who are interested in this method to read these parts carefully. In the past few years, we have successfully applied this method to attack several difficult problems. In a joint work with J.H. Kim [KV2], the method was used in combination with the semi-random method to solve a long standing open problem in finite geometry. In [Vu4], we apply the method to solve an open question of Nathanson on combinatorial number theory. Several other applications can be found in [KV, Vu2, Vu3, Vu5]. Although the proof of Theorem 1.2.1 is fairly technical, it has a decisive advantage that the ideas used in this proof are easily extended to prove the more general and powerful Theorem 1.2.2. In particular, the hardest part of the proof, the concentration arguments, can be generalized without any difficulty. As already discussed, we need to terminate the nibble process once the degree of the hypergraph is more or less equal to its codegree. The key observation here is that when we have proper bounds on the higher degrees, we can prove that the codegree also decreases after each nibble. Thus, we can run the process longer, resulting in a smaller final hypergraph and a better bound on U(H). The statement in Theorem 1.2.2 is obtained by iterating this argument. In order to control the 2-codegree, one has to use a bound on the 3-codegree, and so one has to stop the process before the 2-codegree and the 3-codegree match. On the other hand, one can also prove that the 3-codegree decreases, using information about the 4-codegree and so on. The iteration ends once we reach the s-codegree. We cannot prove that this codegree decreases since we have no information about the s + 1-codegree. Thus, we have to use the original bound Ds for the s-codegree in every step. This requires that the ratio Ds−1 /Ds be (typically) significantly larger than the ratios Dj /Dj+1 , j < s − 1 which explains the difference between assumptions (3) and (4) of Theorem 1.2.2. Now let us take a quick glance at the behavior of the hypergraph in the secondary process. Each bite in the secondary process reduces the vertex set of the hypergraph by a factor of e, i.e., |V (Hi+1 )| ≈ |V (Hi )|/e. Thinking of V (Hi+1 ) as a random subset of V (Hi ), we may expect that the j-codegree is reduced by a factor of ek−j+1 . Therefore, at the ith bite, we may expect that |V (Hi )| ≈ ne−i dv ≈ De−ki for all v ∈ V (Hi ) Cj (Hi ) ≤ Dj ek−j+1 , 2 ≤ j < s. The assumptions of Theorem 1.2.2 will guarantee that when x/ei is sufficiently large (≥ logc D for some constant c), then we can still control the process and prove that Hi ’s −1 ˜ behave as expected. Thus, at the final step we can bound |V (Hf inal )| by O(nx ). Since the major part of uncovered vertices is from V (Hf inal ), this proves the conclusion of Theorem 1.2.2. §2 TOOLS

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Lov´ asz Local Lemma. Consider a set of events A1 , . . . , Am . The dependency graph of A1 , . . . , Am is a graph on {1, . . . , m} such that Ai is mutually independent of all events Aj where i is not adjacent to j. Let di be the degree of i. The following Lemma is a version of the famous Lov´asz Local Lemma [Lo] [AS]. Lemma 2.1. If P r(Ai ) ≤ p and di p ≤ 1/4 for all i, then m P r(∧m i=1 Ai ) ≥ (1 − 2p) .

A Talagrand’s type inequality. Consider the product space Ω generalized by m independent variables x1 , . . . , xm . A function X = X(x1 , . . . , xm ) is called r-Lipschitz if changing the value of any variable changes the value of X by at most r. Let f be a function from N to N, we say that X is f -certifiable if whenever X(x) ≥ b there exists an index set I of at most f (b) elements so that all y ∈ Ω that agrees with x on the coordinates I have X(y) ≥ b. Moreover, for a positive number l, we say that X is l-checkable if X is f -certifiable for some function f such that f (b) ≤ lb for all b. The following inequality is derived by Janson, Shamir, Steele and Spencer [Spe2] from a powerful theorem of Talagrand [Tal]. Talagrand’s inequality. If X is 1-Lipschitz, f -certifiable then for any b and t P r(X ≤ b − t(f (b))1/2 )P r(X ≥ b) ≤ exp(−t/4) Set b = M (X) where M (X) is the median of X, the inequality above implies the following: Claim. If X is r-Lipschitz, 2-checkable then for any t > 0

(2.1)

P r(|X − M (X)| ≥ t(2M (X))1/2 ) ≤ 4exp(−t2 /4r2 ).

The constant 2 in 2M (X) comes from the fact that X is 2-checkable, the constant 4 on the right hand side comes from symmetry and the fact that P r(X ≥ b) = 1/2. Inequality (2.1) gives a strong concentration of X around its median. However, the median of a random variable is usually notoriously hard to compute. Thus it should be better to have a concentration result with respect to the expectation. The following lemma, derives from (2.1), gives such result. We say that X is S-bounded if |X(x)| ≤ S for any x ∈ Ω. Lemma 2.2. Assume that X is an r-Lipschitz, 2-checkable, S-bounded non-negative function. Suppose λ > 0 is chosen so that exp(−λ2 /100r2 E(X))S ≤ λ/10. Then P r(|X − E(X)| ≥ λ) ≤ 4exp(−

λ2 ). 100r2 E(X)

Moreover, P r(X − E(X) ≥ λ) ≤ 2exp(−

λ2 ). 100r2 E(X)

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P r(X − E(X) ≤ −λ) ≤ 2exp(−

λ2 ). 100r2 E(X)

This lemma is tailored for later applications. The condition that X is 2-checkable is not special; one can replace 2 by any positive constant and derive a similar result by the same argument. Moreover, the constant 100 could be improved by tightening the computation. The basic idea of the proof of the lemma is simple: If X is strongly concentrated around its median, then its expectation should be close to the median and thus X is also strongly concentrated around its expectation. λ Proof. Set t = 5E(X) 1/2 . We have (2.2)

4exp(−

t2 λ2 )S = 4exp(− )S ≤ λ/10 4r2 100r2 E(X)

by the assumption. Since X is non-negative 0 ≤ M (X) ≤ 2E(X), by (2.2) and the definition of t we have

(2.3)

t(2M (X))1/2 + 4exp(−

t2 )S ≤ 2t(E(X))1/2 + λ/10 = λ/2. 4r2

Observing that |E(X) − M (X)| ≤ E(|X − M (X)|), we can use (2.1) and (2.3) to show

(2.4)

|E(X) − M (X)| ≤ t(2M (X))1/2 + 4exp(−

t2 )S ≤ λ/2 ≤ 3λ/5. 4r2

Finally, by (2.4) and (2.1) and the fact that M (X) ≤ 2E(X) P r(|X − E(X)| ≥ λ) ≤ P r(|X − M (X)| ≥ λ − |E(X) − M (X)|) ≤ P r(|X − M (X)| ≥ 2λ/5) ≤ 4exp(− ≤ 4exp(−

λ2 (5/2)2 4r2 (2M (X))

)

λ2 ). 100r2 E(X)

The proofs of the remaining two inequalities are similar. ¤ Difficulty with big r. Lemma 2.2 and it preceding two results are very powerful when r is small. Unfortunately, once r is large, they become useless. Suppose that we want to have a large deviation bound for the typical case λ = o(E(X)). Then if r ≥ (E(X))1/2 , both Lemma 2.2 and inequality (2.1) do not provide any information, since the deviation bounds become larger than 1. This fact motivated us to study large deviation bounds for functions with large Lipschitz coefficients. It turns out that if X is a positive polynomial of moderate degree in x1 , . . . , xm , then the situation improves. We could show that for such function X, we can replace the crucial term r2 in the exponent of the bound in Lemma 2.2 by a much smaller quantity.

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

11

This provides a strong concentration result even when r is large. This is the core of our polynomial method and the subject of the rest of this section. Concentration of low degree polynomials. The concentration of low degree polynomials in a product space was systematically investigated in [Vu1], motivated by an earlier result in [KV]. In particular, the result we quote here (Theorem 2.3.) is one of the main results of [Vu1]. Before stating this result, let us introduce some notations. Given a polynomial Y = Y (x1 , . . . , xm ) of degree k and an index set A = {i1 , . . . , ij } we denote by YA the derivative of Y according to the indices in A: YA = ∂ j Y /∂xi1 . . . ∂xij . For all 0 ≤ j ≤ k, Ej (Y ) = maxA,|A|=j E(YA ) (the maximum is taken over all index sets of size j). In particular E0 (Y ) = E(Y ). Example. Let Y = x21 x22 x3 and A1 = {1, 2}, A2 = {1, 1, 3}. Then YA1 (Y ) = 4x1 x2 x3 and YA2 (Y ) = 2x22 . If the expectation of xi is pi , then EA1 = 4p1 p2 p3 and EA2 = 2p22 . Theorem 2.3. There are positive constants ck such that the following holds. Let Y be a positive polynomial of degree k and assume that positive numbers Ej ( 0 ≤ j ≤ k) and λ Ej satisfy: Ej ≥ Ej (Y ) (0 ≤ j ≤ k) and Ej+1 > λ + (j + 1) log m (0 ≤ j ≤ k − 1). Then P r(|Y − E(Y )| > ck λ1/2 (E0 E1 )1/2 ) < 3k e−λ . Theorem 2.3 implies that if the expectation of any partial derivative of Y is much smaller than the expectation of Y , then Y is strongly concentrated around its expectation. The key point here is that the expectation of a partial derivative of Y is usually much smaller than the Lipschitz coefficient of Y . Thus, Theorem 2.3 still gives a good bound in cases where the Lipschitz coefficient is too large for a Talagrand’s type inequality. This is exactly the situation we have to handle in our proof, as we shall point out in the last paragraph of §4.2. Let us illustrate the applicability of Theorem 2.3 by the following example. Assume that tij , 1 ≤ i < j ≤ m be i.i.d. {0, 1} random variables with expectation p = n−2/3 and consider the following function Y =

X

ti1 i2 ti2 i3 ti3 i4 ti4 i1 .

1≤i1 ,i2 ,i3 ,i4 ≤m

It is trivial that the effect of a variable tij can be as large as Θ(m2 ). In other words, Y is Θ(m2 )-Lipschitz. This rules out the chance of using a Talagrand type inequality if E(Y ) = Θ(m4/3 ) ¿ m2 . On the other hand, a partial derivative of order 1 of Y has expectation O(m2 p3 ) = O(1), a partial derivative of order 2 has expectation O(mp2 ) = o(1), and a partial derivative of order 3 or 4 has expectation O(1). It is trivial that each of these quantities is much smaller (4−i)/4 than m2 . Set E0 = E(Y ), Ei = cE0 and λ = m² (where c is a properly chosen constant and 0 < ² < 1/3), we have that for some constant a ²

P r(|Y − E(Y )| ≥ am7/6+²/2 ) ≤ 81e−m . The reader who is familiar with the theory of random graph may have already recognized that Y is nothing other than (8 times) the number of circles of length 4 in the random graph G(m, p). Although the problem of counting small subgraphs of a random graph is classical,

12

VAN. H VU

we do not known how to obtain the above concentration result by any other method (see [Vu1] for more detail). The proof of Theorem 2.3 in [Vu1] shows that we can set c1 ≤ 10 and c2 < 104 (these constants are not optimal, but we do not really care). We now use Theorem 2.3 to derive the following corollaries. Corollary 2.4. Let Y be a polynomial of degree 2. Assume that E0 (Y )/Ej (Y )¢ > 10 log2 m ¡ for j = 1, 2 and E0 (Y ) ≥ E(Y ). Then for λ = 1018 E0 (Y )/ maxj=1,2 Ej (Y ))1/2 , we have P r(Y ≥ 2E0 (Y )) ≤ 9exp(−λ). Proof. Let E0 = maxj=1,2 Ej (Y ). Thus, λ = E2 = E0 , E1 = 108 λE0 ; we have

1 0 1/2 108 (E0 /E )

and E0 = 1016 λ2 E0 . Set

E0 /E1 = E1 /E2 = 108 λ > λ + 3 log m. Moreover, Ej ≥ Ej (Y ) for j = 1, 2. Now by applying Theorem 2.3 for k = 2 we have P r(|Y − E(Y )| > 104 λ1/2 (E0 E1 )1/2 ) ≤ 9exp(−λ). Observe that 104 λ1/2 (E0 E1 )1/2 = E0 ≥ E(Y ), hence P r(Y ≥ 2E0 ) ≤ 9exp(−λ), which ends the proof.

¤ Pm

Corollary 2.5. Let Y = j=1 αi xi , where 0 ≤ αi ≤ r and xi are independent {0, 1} random variables. Assume that E(Y ) ≥ r × 4 log m, then for any ν ≤ E(Y ) P r(|Y − E(Y )| ≥ ν) ≤ 3exp(−

ν2 ). 100E(Y )r 2

ν Proof. Set E0 = E(Y ) and E1 = r. Furthermore, let λ = 100E . Since c1 ≤ 10, it 0 E1 1/2 1/2 follows that c1 λ (E0 E1 ) ≤ ν. Moreover, the assumptions of the lemma guarantee that E0 /E1 > λ + log m. So we can apply Theorem 2.3 to obtain

P r(|Y − E(Y )| ≥ ν) ≤ P r(|Y − E(Y )| ≥ c1 λ1/2 (E0 E1 )1/2 ) ≤ 3exp(−λ) = 3exp(−

ν2 ). 100E(Y )r ¤

§3 BITES Set δk = 1 for k > 3 and δ3 = (30) log3/2 D. Let ²0 = (δk C/D)1/k . For i = 0, 1, . . . , let ni = ne−i , ²i = ²0 ei , di = De−ki . For a quantity f , O∗ (f ) denotes a quantity not exceeding (10k)5 f . We will also use special notations n0i and d0i which represent a quantity of order ¢ Qi ¡ Qi ni j=0 1 + O∗ (²j ) and di j=0 (1 − O∗ (²j )), respectively. If ²i ≤ (10k)−6 , then by the definition of O∗ 20

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

(3.1)

1/2 ≤

i Y ¡

13

¢ 1 ± O∗ (²j ) ≤ 3/2.

j=0

The reader would notice that the constants 1/2 and 3/2 in (3.1) are not the best possible. In fact, most constants in this paper are rather adhoc and fairly loose, so the reader should not worry about the tightness of certain estimates. Big constants such as (10k)5 , (10k)20 , etc could be substantially reduced by tightening the computation. However, as we concentrate only on the order of magnitude, we make no attempt to optimize these constants. Definition. A hypergraph is (m, d, ²)-regular if it has m vertices and for each vertex v, its degree dv is between d(1 − ²) and d(1 + ²). Start with H0 = H which is (n, D, ²0 )-regular (n = n0 , D = d0 by definition). Set T 1/k −T to be the largest integer satisfying ( D e ≥ (10k)20 log D. It is trivial to see that C) D 1/k −20 T = O(log C ). Moreover, ²T ≤ (δk ) (10k) (log D)−1 ≤ (10k)−6 which guarantees that (3.1) holds. Now we are ready to state the Bite lemma; in the rest of this section we let Vi denote the vertex set of the hypergraph Hi . Bite Lemma. For i ≤ T , let Hi be a (m, d, ²i )-regular hypergraph with codegree at most C, where ne−2i ≤ m ≤ n0i and d0i ≤ d ≤ di . Then there is a matching Mi and a set Wi of O(²i ni ) vertices so that the hypergraph Hi+1 induced by Vi+1 = Vi \(V (Mi ) ∪ Wi ) is (m0 , d0 , ²i+1 )-regular for some ne−2(i+1) ≤ m0 ≤ n0i+1 and d0i+1 ≤ d0 ≤ di+1 . We next show that the Bite Lemma implies Theorem 1.2.1. Apply the Bite Lemma for i = 0, 1, . . . , T ; we end up with a sequence of hypergraphs H0 , H1 , . . . , HT +1 , a sequence of matchings M0 , M1 , . . . , MT , and a sequence of vertex sets W0 , W1 , . . . , WT . Observe that the disjoint union M = ∪Ti=1 Mi is a matching of H; moreover, the set of vertices uncovered by M is contained in ∪Ti=0 Wi ∪ VT +1 (we do not have equality here since PT Mi and Wi are not necessarily disjoint). Now, we only have to show that i=1 |Wi | + ˜ 0 n). Notice that by the Bite Lemma, |Wi | = O(²i ni ) = O(²0 n) for all i. |VT +1 | = O(² Therefore, T X

|Wi | = O(²0 n)T = O(²0 n log(

i=1

D )). C

On the other hand, again by the Bite Lemma and (3.1) |VT +1 | ≤ n0T +1 = nT +1

T Y

(1 + O∗ (²i )) = O(nT +1 ).

i=0

Taking into account the definition of ni and T , we have nT +1 = O(ne−T −1 ) = O(n( which completes the proof.

C 1/k ) log D), D ¤

14

VAN. H VU

Remark 3.1. Actually, we prove a little bit more than Theorem 1.2.1. Here we do not require that the original hypergraph H to be D-regular, but only (n, D, ²0 )-regular. So all we need is that for any vertex v D(1 − ²0 ) ≤ dv ≤ D(1 + ²0 ). By reducing ²0 , we can have that |W | ¿ |V (HT +1 )|. This supports an assertion in §1.3 which says that the major part of the set of uncovered points is the vertex set of the hypergraph remaining after the final bite. §4 NIBBLES §4.1 The Nibble Lemma This section deals with the primary processes and shows how a bite can be achieved as the union of several nibbles. Bite Lemma is a consequence of the following. Nibble Lemma. Assume that F is an (m, d, ²)-regular hypergraph with codegree at most ² ² C. Assume further that for any 100k 2 ≤ θ ≤ 30k 2 the following two conditions are satisfied

θ3 m > 104 k 2 min(θ3 (d/100C),

1 d 1 θ( )1/2 ) ≥ 5000k 2 (log + log d). 108 C θ

Then with any θ as above, there are two numbers m0 and d0 , a matching M and a set W of vertices, where

m(1 − 1.9θ) ≤ m0 ≤ m(1 − θ + 4θ²) d(1 − kθ − 3kθ²) ≤ d0 ≤ d(1 − kθ) |W| = O(θ²m) so that the hypergraph F 0 induced by V 0 = V (F )\(V (M) ∪ W) is (m0 , d0 , ²0 )-regular for ²0 = ²(1 + θ). Now we prove that the Nibble Lemma implies the Bite Lemma. This proof is similar to the proof given in the previous section, but the calculation is a little more tedious. Start with F0 = Hi which is (m, d, ²i )-regular (for any i = 0, 1, . . . , T ), where m ≤ n0i and d ≤ di . Set θ = ²i /30k 2 and assume (without loss of generality) that θ−1 is an integer. Apply the Nibble Lemma s = θ−1 times, we end up with a sequence of hypergraphs F0 , F1 , . . . , Fs , a sequence of matchings M0 , M1 , . . . , Ms−1 and a sequence of vertex sets W0 , W1 , . . . , Ws−1 . s−1 Set Hi+1 = Fs , Mi = ∪s−1 j=0 Mj and Wi = ∪j=0 Wj , we first show that Hi+1 , Mi and Wi satisfy the description of the Bite Lemma. To begin, we need to show that Hi+1 is (q, r, δ)-regular, where ne−2(i+1) ≤ q ≤ n0i+1 , 0 di+1 ≤ r ≤ di+1 and δ ≤ ²i+1 . By the Nibble Lemma and the definitions of the related quantities (such as ni , n0i , di , d0i , ²i )

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

q = |V (Fs )| ≤ m(1 − θ + 4(θ²i ))θ q ≥ m(1 − 1.9θ)θ r ≤ d(1 − kθ)θ

−1

−1

−1

≤ n0i e−1 (1 + 4²i ) ≤ n0i+1

≥ me−2 ≥ ne−2i e−2 = ne−2(i+1)

≤ di (1 − kθ)θ

r ≥ d(1 − kθ − 3kθ²i )θ δ ≤ ²i (1 + θ)θ

−1

15

−1

−1

≤ di e−k = di+1

≥ d0i e−k (1 − 4k²i ) = d0i+1

≤ ²i e = ²i+1 ,

which verifies the regularity of Hi+1 . In order to verify the order of magnitude of Wi , let us observe that |Wi | ≤

s−1 X

|Wj | = θ−1 O(θ²i m) = O(²i ni ).

j=0

To complete the proof, we need to show that the following two assumptions of the Nibble Lemma θ3 m > 104 k 2 min(θ3 (d/100C),

1 d 1 θ( )1/2 ) ≥ 5000k 2 (log + log d). 8 10 C θ

are satisfied for all hypergraphs F0 , F1 , . . . Fs−1 . Although the detailed computation is a little bit lengthy, the proof is completely straightforward. To start, let us mention that we use the same θ = ²i /30k 2 for all Fj , j = 0, 1, . . . s. At an intermediate step j, we deal with the hypergraph Fj which is (m[j] , d[j] , ²[j] )-regular. Although we do not know the ratio θ/²[j] exactly, we can always assume that it is between 1/100k 2 and 1/30k 2 (satisfying the condition of the Nibble Lemma), using the fact that ²i = ²[0] ≤ ²[j] ≤ e²[0] = e²i . This is one of the reasons to introduce the secondary process. Without it we have to change θ at every nibble step and this makes the analysis much more technical. The first assumption of the Nibble Lemma is easy to check using the fact that the vertex set does not decrease too fast after each nibble (m0 ≥ m(1 − 2θ)). Notice that at the beginning m[0] = |V (F0 )| = |V (Hi )| ≥ e−2i n and at a generic step j m[j] = |V (Fj )| ≥ −1 (1 − 1.9θ)θ ≥ e−2 m[0] . So it suffices to show that θ3 ne−2i e−2 ≥ 104 k 2 . Since θ > ²i /100k 2 θ3 ≥ (²i /100k 2 )3 = (

δk C 3/k 3i ) e /106 k 6 . D

Therefore

(4.1.1)

θ3 ne−2i e−2 ≥

n(δk C/D)3/k ei−2 . 106 k 6

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VAN. H VU

A simple double counting argument gives that C ≥ D/n, thus the right hand side of 3/k (4.1.1) is at least (δk n1−3/k )/(e2 106 k 6 ) = α. If k ≥ 4, simply set n large enough so that 4 2 α ≥ 10 k . In the case k = 3, α = δ3 /(e2 106 k 6 ) ≥ 104 k 2 due to the definition of δ3 . Remark. Taking a close look at (4.1.1), one can see that in fact we could prove a stronger bound that θ3 m ≥ 104 k 2 log D

(4.1.2)

at every nibble step. Indeed, if k = 3, then (4.1.2) follows from the fact that δ3 = (30)2 0 log3/2 D. Moreover, if k ≥ 4, then the left hand side of (4.1.1) is at least n1/k /e2 106 k 6 = ω(log D) since n ≥ D1/k . We do not need (4.1.2) in the proof of the Nibble Lemma. However, it will be useful in showing the algorithmic efficiency of the proof (see the remark at the end of §4.3). To verify the second assumption, we first prove that F0 = Hi satisfies this assumption with plenty of room to spare. This will imply that the assumption holds for all Fj , due to the fact that d[j] differs from d[0] by only a small constant factor. Recall that Hi is (m, d, ²i )-regular, where d ≥ d0i > 12 De−i > 12 De−T and ²i = (C/D)1/k ei . To show that F0 = Hi satisfies the second assumption of the lemma we need to prove that min(θ3 (d/100C),

1 d θ( )1/2 ) ≥ 5000k 2 (log θ−1 + log d) 8 10 C

where ²i /100k 2 ≤ θ ≤ ²i /30k 2 . Since both θ−1 and d are upper bounded by D, it suffices to prove that

(4.1.3)

min(θ3 (d/100C),

1 d θ( )1/2 ) ≥ 104 k 2 log D. 8 10 C

Consider a = θ3 d/100C. Since d ≥ di /2 and θ ≥ ²i /100k 2 , a ≥ ²3i di /(2 × 108 k 6 C). Recalling that ²i = (δk C/D)1/k ei and di = De−ki it follows that 3/k

a≥

((D/C)1/k e−i )k−3 δk 2 × 108 k 6

.

If k ≥ 4, δk = 1 and k − 3 ≥ 1, so (D/C)1/k e−i (D/C)1/k e−T ≥ 2 × 108 k 6 2 × 108 k 6 20 (10k) log D ≥ À 104 k 2 log D 2 × 108 k 6

a≥

by the definition of T (T is chosen so that (D/C)1/k e−T ≥ (10k)20 log D). If k = 3 then by the definition of δ3 a≥

(30)20 (log D)3/2 δ3 ≥ À 104 k 2 log D. 2 × 108 k 6 2 × 108 k 2

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

17

Consider b = 1018 θ( Cd )1/2 . We need to show that b2 ≥ 108 k 4 (log D)2 . Taking into account that θ > ²/100k 2 = ²i /100k 2 and d ≥ di /2, it suffices to show that (

1 1 1 ²2i di )( )( )( ) ≥ 108 k 4 (log D)2 , 1016 104 k 4 2 C

which is equivalent to 2/k

((D/C)1/k e−i )k−2 δk

≥ 2 × 1028 k 8 (log D)2 .

If k ≥ 4 then δk = 1 and again by the definition of T ((D/C)1/k e−i )k−2 ≥ ((D/C)1/k e−T )2 À ((10k)20 log D)2 À 1028 k 8 log2 D. If k = 3 then 2/3

((D/C)1/3 e−i )3−2 δ3

2/3

≥ (D/C)1/3 e−T δ3

≥ (30)20 log D((30)20 (log D)3/2 )2/3 À 2 × 1028 38 (log D)2 . This completes (with a lot of room to spare) the proof for F0 . To show the same thing holds for Fj , j ≥ 1, notice that d[j] ≥ d[s] > d[0] /10 = di /10 and ²[j] ≥ ²[0] = ²i for all j. The factor 1/10 is swallowed by the huge gap between what we actually shown for j = 0 and the required bound. ¤ §4.2 Proof of the Nibble Lemma Given a hypergraph F and θ as in the Nibble Lemma, we define M and W as the outcome of a random process with inputs F and θ. We shall show that with positive probability, M and W will satisfy the requirements of the Lemma. Let V and E denote the vertex and the edge set of F , respectively. Furthermore, let dmin and dmax be the smallest and the largest degree of F , respectively. Set p = θ/dmin , and for each v ∈ V , set 0 < pv < 1 so that 1 − pv = (1 − p)dmax −dv . It is useful to keep in mind that Ω(²) = θ ¿ ² ¿ 1. Some facts about the parameters. The following facts will be used frequently: (4.2.1) (4.2.2) (4.2.3) (4.2.4) (4.2.5) (4.2.6)

²/100k 2 ≤ θ ≤ ²/30k 2 pv ≤ 3²θ for any v ∈ V dmax /dmin < 1 + 3² < 1.01 < 1.9 < 2 1 d 1 min(θ3 (d/100C), 8 θ( )1/2 ) ≥ 5000k 2 (log + log d) 10 C θ θ3 m ≥ 104 k 2 ²0 = ² + ²θ > ² + 20k 2 θ2 .

Notice that (4.2.4) and (4.2.5) are the assumptions of Nibble Lemma, (4.2.1) is the definition of θ and implies (4.2.6). The other facts follow from the following (1 + ²)d ≥ dmax ≥ d ≥ dmin ≥ (1 − ²)d.

18

VAN. H VU

How to conduct a nibble. For each e ∈ E, choose e with probability p; furthermore, for any v ∈ V choose v with probability pv (all choices are independent). We represent the choice of e by an atom random variable te and the choice of v by an atom variable ρv ; by definition P r(te = 1) = p, P r(te = 0) = 1 − p, P r(ρv = 1) = pv and P r(ρv = 0) = 1 − pv . We say an edge e is lonely if e is chosen and disjoint from all other chosen edges. Let M be the collection of lonely edges; it is trivial that M is a matching. To define W, let E be the set of chosen, but not lonely edges and V0 be the set of chosen vertices. Now let W = V (E) ∪ V0 . It is clear that both M and W are random variables depending on the choices of edges and vertices. We say that a vertex v survives if v ∈ V 0 = V \(V (M) ∪ W), otherwise, v is dead. For any v ∈ V , the degree d0v of v in F 0 is the number of k tuples v = {v1 , . . . , vk }, where all vi ∈ v survive and v ∪ v is an edge in F . (Notice that the degree is defined even if v is dead; when v survives, it is the actual degree of v in F 0 ). Stabilization. The purpose of choosing each point v with (different) probability pv as defined is to make the probability that a vertex survives equal for all vertices. This, in a sense, stabilizes the process and keep the gap between the degrees in F 0 small. A vertex v ∈ V is normal if the degree d0v of v in F 0 is between dv (1 − kpdmax − 15k 2 θ2 ) and dv (1 − kpdmax + 15k 2 θ2 ). Set d0 = d(1 − kpdmax ), using (4.2.3) one can check that d0 satisfies the requirement of the lemma that (1 − kθ − 3kθ²)d ≤ d0 ≤ (1 − kθ)d. Next, we observe that if v is normal, then its degree d0v in F 0 satisfies the requirement of the lemma. Observation. If v is normal, then the degree d0v of v in F 0 satisfies (1 − ²0 )d0 ≤ d0v ≤ (1 + ²0 )d0 . Proof. If v is normal then by definition and (4.2.6) d0v ≤ dv (1 − kpdmax + 15k 2 θ2 ) ≤ d(1 + ²)(1 − kpdmax + 15k 2 θ2 ) d0 (1 + ²)(1 − kpdmax + 15k 2 θ2 ) 1 − kpdmax ≤ d0 (1 + ²)(1 + 16k 2 θ2 ) ≤ d0 (1 + ²0 ).

=

The lower bound could be verified the same way. Consider the following four events: A = {(1 − 1.9θ)m ≤ |V 0 | ≤ m(1 − θ + 4θ²)} B = {|V0 | ≤ 4θ²m} C = {|E| ≤ 5θ2 m} Dv = { v is normal}.

¤

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

19

It is easy to check that if A, B, C and ∧v∈V Dv hold simultaneously, then M and W satisfy the requirement of the Lemma. Indeed, B and C imply that |W| = |V0 ∪ V (E)| = O(θ²m). Moreover, A provides the bounds on the size of V 0 . Finally, ∧v∈V Dv and the fact just proven imply that H 0 is (|V 0 |, d0 , ²0 )-regular. So it suffices to prove that with positive probability, A, B, C and ∧v∈V Dv hold simultaneously. To do this, it suffices to show

(4.2.7)

P r(A) + P r(B) + P r(C) + P r(∧v∈V Dv ) < 1.

(4.2.7), on the other hand, is a corollary of the following two claims. Claim 4.2.1. P r(A) ≤ 4exp(−θ3 m/1600k 2 ). P r(B) ≤ 2exp(−θ3 m/1600k 2 ). P r(C) ≤ 2exp(−θ3 m/1600k 2 ). Claim 4.2.2. P r(∧v∈V Dv ) ≥ exp(−100me−5000k

2

(log

1 θ +log d)

).

To see that the above two claims yield (4.2.7), we need only show

(4.2.7)

exp(−100me−5000k

2

(log

1 θ +log d)

) ≥ 8exp(−θ3 m/1600k 2 ).

By (4.2.5), θ3 m/1600k 2 ≥ 6. Therefore 8exp(−θ3 m/1600k 2 ) ≤ exp(−θ3 m/800k 2 ). 2

1

Firhtermore, θ3 m/800k 2 À 100me−5000k (log θ +log d) . This and the previous line imply (4.2.7). To finish the proof of the Nibble Lemma, it thus remains to prove Claims 4.2.1 and 4.2.2. The proof of Claim 4.2.1 relies on Lemma 2.2. The quantities in A, B and C (|V 0 |, |V0 | and |E|, respectively) are random variables depending on the atom variables te and ρv (e ∈ E, v ∈ V ). Moreover, each atom variable has small effect on these quantities; thus, we can apply Lemma 2.2 to prove strong concentration results which imply the statements of Claim 4.2.1. The details follow. Proof of Claim 4.2.1. A. The event that a vertex v survives can be written as follows I(v survive) =

Y

(1 − te )(1 − ρv ).

e3v

So the probability that v survives is (1−p)dv (1−pv ) = (1−p)dmax = p0 , by the definition of pv . Since dmax /dmin ≤ 1.01, a simple calculation shows 1 − 1.8θ ≤ p0 ≤ 1 − θ + 3θ². Therefore, m(1 − 1.8θ) ≤ p0 m = E(|V 0 |) ≤ m(1 − θ + 3θ²).

20

VAN. H VU

Let us consider V 00 = V \V 0 , the set of dead vertices; it follows from the bound on E(|V 0 |) that 1.8mθ ≥ E(|V 00 |) ≥ m(θ − 3θ²). The main trick in this proof is that instead of proving a concentration result for |V 0 |, we do it for |V 00 |. To apply Lemma 2.2, it is crucial to show that each atom variable has very small effect on |V 00 |. We claim that changing one atom variable cannot change |V 00 | by more than k + 1. First, observe that changing ρv for any v changes |V 00 | by at most 1. Moreover, changing any te changes |V 00 | by at most the size of one edge, which is k + 1. Since k + 1 ≤ 2k, we can assume, that |V 00 | (as a function depending on the atom variables) is 2k-Lipschitz. Now we investigate the checkability of |V 00 |. To certify that a vertex v is dead, it is enough to show either ρv = 1 or te = 1 for some e 3 v. Thus, to show |V 00 | ≥ s, we need only s coordinates, i.e, |V 00 | is 1-checkable. Since |V 00 | ≤ m, V 00 is m-bounded. Set λ = θ²m; in order to apply Lemma 2.2, we need to show that the condition on λ in Lemma 2.2 is satisfied. Given the properties of |V 00 | (as a random variable), it is enough to show that exp(−

λ2 )m ≤ λ/10, 100(2k)2 E(|V 00 |)

or equivalently,

(4.2.8)

exp(−

θ 2 ² 2 m2 )m ≤ θ²m/10. 100(2k)2 E(|V 00 |)

Since θ ≤ ²/10 and E(|V 00 |) ≤ 2mθ, (4.2.8) follows from exp(−

θ²2 m ) ≤ θ2 , 800k 2

which, in turn, is a trivial corollary of (4.2.4) and the fact that m ≥ Lemma 2.2 we obtain

d C.

Now applying

P r(|V 00 | ≤ m(θ − 4θ²)) ≤ P r(|V 00 | − E(|V 00 |) ≤ −θ²m) θ2 ²2 m2 ) 100(2k)2 E(|V 00 |) θ3 m ≤ 2exp(− ). 1600k 2

≤ 2exp(− (4.2.9)

In the last inequality we use the trivial facts that θ < ² and E(|V 00 |) ≤ 2θm and 800 < 1600. The other bound on |V 00 | can be proven similarly P r(|V 00 | ≥ 1.9mθ) ≤ P r(|V 00 | − E(|V 00 |) ≥ θm/10) (θm/10)2 ) 100(2k)2 E(|V 00 |) θ3 m ≤ 2exp(− ), 1600k 2 ≤ 2exp(−

(4.2.10)

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

21

using θ < 10−2 in the last inequality. Since |V 0 | = m − |V 00 |, (4.2.9) and (4.2.10) together imply the desired bounds on |V 0 | and prove the first statement of Claim 4.2.1. The reader may wonder why we did not apply Lemma 2.2 directly to |V 0 |. The reason is quite fascinating but probably a little bit morbid: it is easier to certify death than life. It requires only one witness to show that a vertex v is dead (the one that kills it, say); but to check whether v survives, one has to ask the whole population to make sure that nobody kills v. B. It is clear that |V0 | isP the sum of the independent random variables ρv (v ∈ V ). Thus, the expectation of |V0 | is v∈V pv ≤ 3θ²m, by (4.2.2). It is clear that |V0 | is 1-checkable and 1-Lipschitz. Setting λ = θ²m, by Lemma 2.2. and the fact that θ < ², we have P r(|V0 | ≥ 4θ²m) ≤ 2exp(−(θ²m)2 /100(3θ²m)) ≤ 2exp(−θ3 m/1600k 2 ). Here we omit the (easy) proof that λ satisfies the condition of Lemma 2.2. C. Recall that an edge e is in E if e is chosen and there is another chosen edge intersecting e. Let δe be the number of edges intersecting e; the probability that none of these edges is chosen is (1 − p)δe ≤ (1 − p)(k+1)dmax . Since pdmax = θdmax /dmin < 2θ by (4.2.3) P r(e ∈ E) ≤ p(1 − (1 − p)(k+1)dmax ) ≤ p2 (k + 1)dmax ≤ 2θ(k + 1)p. P It then follows that the expectation of E is at most 2θ(k +1)p|E| = 2θ(k +1)p v∈V dv /(k + 1) ≤ 4θ2 m. The crucial note here is that changing any atom variable te would change E by at most k + 1 ≤ 2k since all chosen edges not in E are disjoint; moreover, ρv has no effect on E. So |E| is (2k)-Lipschitz. To show that e ∈ E, it suffices to have te = tf = 1 for some f intersecting e. Thus to certify that |E| ≥ s, it is enough to look at 2s coordinates. This means that |E| is 2-checkable. Setting λ = θ2 m and applying Lemma 2.2, we have P r(|E| ≥ 5θ2 m) ≤ 2exp(−(θ2 m)2 /100(2k)2 (4θ2 m)) = 2exp(−θ2 m/1600k 2 ) ≤ 2exp(−θ3 m/1600k 2 ). Here again we omit the easy proof of the fact that λ satisfies the condition of Lemma 2.2. The proof of Claim 4.2.1 is now complete. ¤ Claim 4.2.2 is a corollary of the following claim and the Lov´asz Local Lemma. Claim 4.2.3. For any v ∈ V , P r(Dv ) ≤ 30exp(−5000k 2 (log

1 θ

+ log d)).

Assume for a moment that Claim 4.2.3. holds. Claim 4.2.2 can be derived as follows. First notice that the degree of any vertex v is determined by the edges that intersect with at least one edge containing v. So, if the distance between v and v 0 in the hypergraph is at least 4, then any choice that effects the degree of v does not effect the degree of v 0 . In other words, the degrees of v and v 0 are independent. Consequently, the dependency degree of Dv is at most the number of points with distance 3 from some vertex. This number is upper bounded by kdmax + k 2 d2max + k 3 d3max ≤ 2k 3 d3 . Since 30exp(−5000k 2 (log θ1 + log d)) × 2k 3 d3 < 1/4, we can apply the Local Lemma to estimate P r(∧v∈V Dv ) P r(∧v∈V Dv ) ≥ (1 − 60e−5000k

2

(log

≥ exp(−100me−5000k

1 θ +log d) 2

(log

)m

1 θ +log d)

).

22

VAN. H VU

The only thing that remains is to prove Claim 4.2.3. However, the proof of this claim is significantly more complicated than that of Claim 4.2.1. The principal obstruction here is that the atom variables (te ’s and ρv ’s) could have a very strong effect on the degree of a vertex–in other words, the function in question has huge Lipschitz coefficient. Fix a vertex v and, for instance, change an atom variable ρu from 0 to 1. Let Xv = d0v denote the degree of v in H 0 . The change of ρu may decrease Xv by as much as the number of edges containing both v and u. Since this number can be as large as C, X is, at best, C-Lipschitz. Although it is not clear how to certify X, we assume, for simplicity that X is 1-checkable (in general one cannot hope for more than this). Let us now see what we can obtain from Lemma 2.2. under the assumption that X is C-Lipschitz and 1-checkable. Since the expectation of d0v is roughly d(1 − kθ), by setting r = C Lemma 2.2. (if applicable) yields

(4.2.11)

P r(|Xv − E(Xv )| ≥ Ω(θ2 d)) ≤ exp(−(θ2 d)2 /Ω(C 2 d)) = exp(−O(θ4 d/C 2 )).

The trouble is the term θ4 d/C 2 in the exponent. It is clear that when d = O(C 2 ) this term is of order o(1), and the deviation bound obtained in (4.2.11) is useless. This means that by (4.2.11) we cannot continue the nibble process further once the degrees of the hypergraph are of order O(C 2 ). On the other hand, in order to obtain our result, we ˜ should continue until the degrees get down to O(C). One may also try to use the querygame martingale method from [AKS], but it seems to us that this method also leads to the same problem. The situation calls for a new large deviation method which can be applied to Claim 4.2.3. The so-called polynomial method described in §1.3 turns out to be the perfect tool for the case, as one will see in the next section. §4.3 Proof of Claim 4.2.3 Fix a point v ∈ V and let Xv be the degree of v in F 0 . The key observation of the proof is that Xv can be expressed as a polynomial in the atom variables X Y Y Xv = ( (1 − tf )(1 − ρu )). e3v u∈e\v f 3u

In the rest of the proof, we omit the unnecessary subindex v (as v is fixed),Q and write X q for Xv . Let y = (y1 , . . . , yq ) be a vector of length q and h(y) = h(y1 , . . . , yq ) = j=1 (1−yj ). Pq Pq P We call h1 (y) = 1 − j=1 yj and h2 (y) = 1 − j=1 yj + 1≤j6=j 0 ≤q yj yj 0 the first and second order approximation of h, respectively. The following observation is trivial, but very important: For any {0, 1} vector y h1 (y) ≤ h(y) ≤ h2 (y). Q Set he = u∈e\v ( f 3u (1 − tf )(1 − ρu )). Since tf and ρu are {0, 1} random variables, by the above observation, we have X X X he2 = X2 . he ≤ he1 ≤ X = X1 = Q

e3v

e3v

e3v

Claim 4.2.3. is a straightforward consequence of the following two facts and (4.2.4).

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

23

Fact 4.3.1. With probability at least 1 − 3exp(−θ3 d/100C) the following holds dv (1 − kpdmax − 3kθ2 ) ≤ X1 ≤ dv (1 − kpdmax + 4kθ2 ). Fact 4.3.2. With probability at least 1 − 27exp(− 1018 θ( Cd )1/2 ) the following holds X2 − X1 ≤ 10k 2 dv θ2 . Proof of Fact 4.3.1. By definition X 1 = dv −

X X X ( tf + ρu ) = dv − U1 . e3v u∈e\v f 3u

P Consider E( f 3u tf +ρu ) = pdu +pu . To estimate this number, let us recall the definition of pu ; pu is determined so that (1 − p)du (1 − pu ) = (1 − p)dmax . By the Taylor series expansion, it follows that (dmax − du )p ≥ pu ≥ (dmax − du )p − (dmax − du )2 p2 . On the other hand, (dmax − du )2 p2 = θ2 (dmax − du )2 /d2min ≤ θ2 . Thus (4.3.1)

dmax p ≥ pdu + pu ≥ dmax p − θ2 .

Therefore,

(4.3.2)

dv kdmax p ≥ E(U1 ) ≥ dv (kdmax p − kθ2 ).

Consider a vertex u, u can appears in at most C edges containing v because of the codegree condition. Thus each variable ρu can be repeated in U1 at most C times. Similarly, each tf is repeated in U1 at most (k + 1)C ≤ 2kC times. Thus, U1 is a linear function in term of the atom variables with coefficients at most 2kC. Setting ν = 3kθ2 dv , Corollary 2.5 implies

(4.3.3)

P r(|U1 − E(U1 )| ≥ 3kθ2 dv ) ≤ 3exp(−(3kθ2 dv )2 /100E(U1 )2kC).

Since E(U1 ) ≤ 2kdv θ and dv ≥ 0.9d,

(4.3.4)

3exp(−(3θ2 kdv )2 /100E(U1 )2kC) ≤ 3exp(−θ3 d/100C).

Combining (4.3.2) (4.3.3) and (4.3.4) finishes the proof. Proof of Fact 4.3.2. We can write X2 − X1 = Y1 + Y2 + Y3 , where

¤

24

VAN. H VU

Y1 =

X

X

X

tf tf 0

e3v u,u0 ∈e\v f 3u,f 0 3u0

Y2 =

X

X

X

t f ρu 0

e3v u,u0 ∈e\v f 3u

Y3 =

X

X

ρu ρu0 .

e3v u6=u0 ∈e\v

We estimate Y1 , Y2 , Y3 separately, using Corollary 2.4. First, consider Y1 . To bound the expectation of Y1 , notice that E(tf tf 0 ) = p2 if f 6= f 0 and E(tf tf 0 ) = p otherwise. The number of pairs (f, f 0 ) in the sum is at most dv k(k − 1)d2max . Moreover, the number of pairs (f, f 0 ) where f and f 0 represent the same edge is at most dv k(k − 1)C. Therefore, (4.3.5)

E(Y1 ) ≤ dv k(k − 1)d2max p2 + dv k(k − 1)Cp ≤ 2θ2 k 2 d + 2θk 2 C ≤ 4k 2 θ2 d.

Fix an edge f , it is not too hard to show that there are at most (k + 1)Ckdmax edges f 0 such that f 6= f 0 and tf tf 0 appears in the sum (there are k + 1 ways to choose a vertex u of f ; for each u there are at most C ways to choose an edge e through u and v; from each such e there are k ways to choose u0 6= v and each u0 has at most dmax edges). Moreover, ¡ ¢ by a similar argument one can show that a term tf tf can appear at most k2 C times (there ¡ ¢ are k2 ways to choose u 6= u0 from f \v; for each pair (u, u0 ) there are at most C edges containing both u, u0 and v). This implies

(4.3.6)

E1 (Y1 ) ≤ (k + 1)Ckdmax p +

µ ¶ k 1 C ≤ 2k 2 θC + k 2 C ≤ k 2 C. 2 2

Finally, to bound E2 (Y1 ), we need to estimate the number of repetitions of tf tf 0 (f 6= f 0 ). Fix such pair (f, f 0 ). There are at most (k + 1)2 way to choose a pair of vertices (u ∈ f, u0 ∈ f 0 ). For any pair (u, u0 ), there are at most C edges containing all three points u, u0 and v. This yields that E2 (Y1 ) ≤ (k + 1)2 C.

(4.3.7) Set E0 = 4k 2 θ2 d and λ = (4.2.4) implies that

E0 1 1/2 . 108 ( (k+1)2 C )

By (4.3.5)–(4.3.7), E0 ≥ E(Y1 ). Moreover,

θ2 d À log2 d. C Notice that the number of variables involved in the polynomial in question is polynomial in d; so, all conditions of Lemma 2.4 are satisfied and we can use this lemma to obtain E0 / max(E1 (Y1 ), E2 (Y2 )) ≥

(4.3.8)

P r(Y1 ≥ 8k 2 θ2 d) ≤ 9exp(−λ) ≤ 9exp(−

1 d θ( )1/2 ). 8 10 C

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

25

The bounds concerning Y2 and Y3 can be derived by a similar and somewhat simpler computation. First consider X X X Y2 = tf ρu0 . e3v u,u0 ∈e\v f 3u

We have by (4.2.2) and (4.2.3) E(Y2 ) ≤ dv k 2 dmax p max pu0 ≤ (2dθ)k 2 (3θ²) = 6dk 2 θ2 ² ¿ k 2 θ2 dv . 0 u ∈V

To bound E1 (Y2 ), observe that for each edge f , there are at most (k + 1)2 C vertices u so that tf ρu0 could occur in the sum. Moreover, for each vertex u0 , there are at most Ckdmax edges f with the same property. Thus, 0

E1 (Y2 ) ≤ max((k + 1)2 C max pu0 , kCdmax p) ≤ max(3(k + 1)2 Cθ², 2kCθ) = 2kCθ ¿ C. 0 u ∈V

For a fixed pair (f, u0 ), the term tf ρ0u can be repeated at most (k + 1)C times. Hence, E2 (Y2 ) ≤ (k + 1)C. Now consider Y3 =

X

X

ρu ρu 0 .

e3v u6=u0 ∈e\v

By the bound on pu in (4.2.2), we have E(Y3 ) ≤ dv

µ ¶ k (3θ²)2 ¿ k 2 θ2 dv . 2

E1 (Y3 ) ≤ (k − 1)C3θ max pu ≤ 3kCθ² ¿ C. u

E2 (Y3 ) ≤ C. For each Yi , set E0 = k 2 θ2 dv /2 and λ = λ=

1 108 (E0 /(k

+ 1)C)1/2 Notice that

1 k2 1 1/2 θ(d /C) ( )1/2 ≥ 8 θ(dv /C)1/2 (9/8)1/2 . v 108 2(k + 1) 10

By Corollary 2.4, it follows (with a lot of room to spare) that for i = 2 and 3

(4.3.9)

P r(Yi ≥ k 2 θ2 dv ) ≤ 9exp(−λ) ≤ 9exp(−

1 θ(d/C)1/2 ). 108

Combining (4.3.8) and (4.3.9), we have P r(X2 − X1 ≥ 10k 2 θ2 dv ) ≤ 27exp(−

1 d θ( )1/2 ), 8 10 C

26

VAN. H VU

which completes the proof. ¤ Remark 4.3.3. Algorithmic efficiency: The proof we present here provides an efficient algorithm to find a large matching in the case D is a positive power of n, namely, D ≥ nα for some positive constant α. For instance, both applications considered in §1.2 are in this category. Let us assume, for simplicity, that α = 0.1. Consider (4.2.7). Since the left hand side of (4.2.7) is an upper bound for the failure probability of one nibble step and there are O(D) nibble steps, it is enough to show that if D ≥ nα , then the left hand side of (4.2.7) is of order O(D−2 ). This would imply that the whole process produces a matching as desired with probability 1 − o(1). To bound the left hand side of (4.2.7), notice that since α = 0.1 (4.1.3) yields

(4.3.10)

min(θ3 (d/100C),

1 d θ( )1/2 ) ≥ 104 log D ≥ 103 log n ≥ 103 log m. 8 10 C

Therefore, it follows by Facts 4.3.1 and 4.3.2 that P r(∧v∈V Dv ) ≤ m × 30e−10

4

log D

¿ e−D .

Now consider the bound in Claim 4.2.1. By (4.1.2) θ3 m ≥ 104 k 2 log D. Thus all probabilities in Claim 4.2.1. are upper bounded by e−2logD = D−2 . Together, the failure probability in (4.2.7) is of order O(D−2 + e−D ) = O(D−2 ), which completes the proof. In fact, in this case the P failure probabilities P r(Dv ) is so small so that we can avoid the Local Lemma, and use v∈V P r(Dv ) to upper bound P r(∧v∈V Dv ). For arbitrary α, to guarantee (4.3.10), simply adjust T to strengthen (4.2.4). ¤ §5 PROOF OF THEOREM 1.2.2 Let ²0 = αx−1 logβ D and T be the largest integer so that e−T ≥ γx−1 log D, where α, β and γ are positive constants to be determined later. Let ²i = ²0 ei and Dj,i = Dj e−(k−j+1)i Qi 0 = Dj,i l=1 (1 + O∗ (²l )) (for the definition of O∗ , for all 0 ≤ i ≤ T . Furthermore, let Dj,i see §3). The definitions of ni , n0i , di , d0i are unchanged. The following lemma is the extended version of the Bite Lemma in §3. 0 Lemma 5.1. Suppose that Hi (i ≤ T ) is (m, d, ²i )-regular and Cj (Hi ) ≤ Dj,i for all 1 < j < s and Cs (Hi ) ≤ Ds . Then there is a matching Mi and a set Wi of O(²i ni ) vertices such that the hypergraph Hi+1 induced by Vi+1 = Vi \(V (Mi ) ∪ Wi ) satisfies the following: (1) Hi+1 is (m0 , d0 , ²i+1 )-regular for some ne−2(i+1) ≤ m0 ≤ n0i+1 and d0i+1 ≤ d0 ≤ di+1 . 0 (2) Cj (Hi+1 ) ≤ Dj,i+1 for all 1 < j < s.

Repeating the argument in § 3, one can see that Lemma 5.1 implies Theorem 1.2.2. To prove Lemma 5.1, we need the following lemma, which is an extension of the Nibble Lemma. Lemma 5.2. Assume that F is an (m, d, ²)-regular hypergraph with Cj (F ) ≤ bj for 1 < j ≤ s. Assume further that for any ²/100k 2 ≤ θ ≤ ²/30k 2 the following conditions are satisfied

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

27

(1) θ3 m > 104 k 2 (2) min(θ3 (d/100b2 ), 1018 θ(d/b2 )1/2 ) ≥ 5000k 2 (log θ1 + log d). (3) For all 1 < j ≤ s − 1, min(θ3 (bj /100bj+1 ), 1018 θ(bj /bj+1 )1/2 ) ≥ 5000k 2 (log θ1 + log d). Then with any θ as above, there are two numbers m0 and d0 , a matching M and a set W of vertices, where m(1 − 1.9θ) ≤ m0 ≤ m(1 − θ + 4θ²) d(1 − kθ − 3kθ²) ≤ d0 ≤ d(1 − kθ) |W| = O(θ²m) so that the hypergraph F 0 induced by V 0 = V (F )\(V (M) ∪ W) is (m0 , d0 , ²0 )-regular for ²0 = ²(1 + θ) and Cj (F 0 ) ≤ bj (1 − (k − j + 1)θ + 15θ2 ) for all 1 < j < s. Again it is not too hard to see that Lemma 5.2 implies Lemma 5.1 (in the way the Nibble Lemma implies the Bite Lemma). However, let us explain why assumptions (3) and (4) of Theorem 1.2.2 are important. Assume that we are dealing with a nibble in the ith bite. Since each bite reduces the size of the vertex set by a factor e, we expect that it reduces the degree by a factor ek and the j-codegree by a factor e(k−j+1) . Thus, we expect that in bite ith

(5.1) (5.2) (5.3)

m = Θ(ni ) = Θ(ne−i ) d = Θ(di ) = Θ(De−ki ) bj = O(Dj,i ) = O(Dj e−(k−j+1)i ), for all 1 < j < s.

Similar to the proof in §4.1, we need to check that the assumptions (1)(2)(3) of Lemma 5.2 are satisfied a nibble. In particular, to verify (2) we have to show

(5.4)

˜ θ3 (d/b2 ) ≥ 104 log D = O(1).

Recall that θ = Θ(²i ) = Θ(²0 ei ). By this and (5.2) and (5.3) the left hand side in (5.4) is ˜ the assumption of order Θ(²30 e2i D/D2 ) which has minimum at i = 0. Since ²0 = x−1 O(1), D/D2 = D1 /D2 ≥ x3 ( along with proper constants α, β, γ in the definition of ²0 and T ) can be used to prove (5.4). Similarly, to verify assumption (3) of Lemma 5.2. we need to show

(5.5)

˜ θ3 (bj /bj+1 ) ≥ 104 log D = O(1).

For j < s − 1, an identical argument will lead to the necessity of the condition Dj /Dj+1 ≥ x3 , which explains the assumption (3) of Theorem 1.2.2. To explain assumption (4), consider the case j = s − 1. We have bs−1 = O(Ds−1 e−(k−s+2)i ) by induction. But since we cannot control codegrees of order higher than s − 1, we can only say bs = Ds (instead of bs = O(Ds e−(k−s+1)i )). Thus the ratio bs−1 /bs is different from all other ratios bj /bj+1

28

VAN. H VU

(5.6)

θ3 bs−1 /bs = Θ(²30 e3i Ds−1 e−(k−s+2)i /Ds ).

The right hand side of (5.6) has minimum at either i = 0 or i = T . If the minimum is at i = 0, we can again use assumption (3). If the minimum is obtained at T its value will be −(k−s−1)T ˜ Θ(x−3 O(1)e Ds−1 /Ds ) = a.

˜ ˜ a = Θ(x−(k−s+2) O(1)D Since e−(k−s−1)T = x−(k−s−1) O(1), s−1 /Ds ). Assumption (4) of the theorem is provided in order to guarantee that the last quantity is sufficiently large to satisfy (5.5). To finish the proof, it remains to prove Lemma 5.2. The difference between this lemma and the Nibble Lemma is that here we have one more assumption (assumption (3)) along with one more requirement (the bounds on Cj (F 0 )’s). We shall prove Lemma 5.2 by considering the same nibble and showing that with the additional assumption (3), the probability that the output F 0 satisfies both the original and the additional requirements is positive. The proof is generally similar, but somewhat more delicate than the proof of the Nibble Lemma. In particular, the following definition will be essential. We say a set S of s − 1 vertices is important if codeg(S) > 0 and let S be the set of all important S’s. For any S ∈ S, we say that S is perfect if the following holds • For every vertex v ∈ S, v is normal, namely, the degree d0v of v in F 0 satisfies dv (1 − kpdmax − 15k 2 θ2 ) ≤ d0v ≤ (1 − kpdmax + 15k 2 θ2 )dv . • For every subset v of S, |v| = j > 1, codeg 0 (v) ≤ bj (1 − (k − j + 1)pdmax + 15k 2 θ2 ). Here codeg 0 (v) is defined similarly to d0v . It is the number of l (= k + 1 − |v|) tuples (v1 , . . . , vl ) such that all vi survives and v ∪ {v1 , . . . , vl } is an edge of F . It is obvious but crucial that • Every vertex is contained in some important set. • Every set v with at most s − 1 vertices with codegree at least 1 is contained in some important set. Now consider events A, B, C and DS where A, B and C are defined as in §4.2 and DS = {S perfect}. Similar to the argument prior to (4.2.7), we can see that if A, B, C and ∧S∈S DS hold simultaneously, then the requirements of Lemma 5.2. are satisfied. Now, instead of (4.2.7), we need to prove the following

(5.7)

P r(A) + P r(B) + P r(C) + P r(∧S∈S DS ) < 1.

To show this, it suffices to prove

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

Claim 5.3. P r(∧S∈S DS ) ≥ exp(−200(4k)k mde−5000k

2

(log

1 θ +log d)

29

).

Claim 5.3 and Claim 4.2.1 imply (5.7) in the similar way Claim 4.2.2 and Claim 4.2.1 imply (4.2.7). It remains to verify Claim 5.3. Similar to Claim 4.2.2, Claim 5.3 can be deduced from the Local Lemma and the following Claim 5.4. For any S ∈ S, P r(DS ) ≤ 30 × 2k exp(−5000k 2 (log

1 + log d)). θ

However, the application of the Local Lemma here is more delicate than in the proof of Claim 4.2. First we need the following fact Fact 5.5. • |S| ≤ 2md(k + 1)s−2 . • In the dependency graph of the events {S is perfect} the maximal degree is at most (2k 4 d4 )k . Proof. The first statement follows by a simple double counting argument. For each set S ∈ S, by¡ definition there is an edge eS which contains S. Since each edge e can be counted ¢ at most k+1 times and there are at most (1 + ²)dm/(k + 1) ≤ 2dm/(k + 1) edges, we can s−1 conclude that µ ¶ k+1 |S| ≤ 2dm /(k + 1) ≤ 2dm(k + 1)s−2 . s−1 To prove the second statement, notice that if the events corresponding to S and S 0 are dependent, then there should be a vertex v ∈ S and a vertex v ∈ S 0 such that the distance between v and v 0 are at most 3. On the other hand, all vertices in S 0 are adjacent to v 0 , thus all vertices in S 0 are of distance at most 4 from v. Since there are less than 2k 4 d4 vertices of distance at most 4 from v, the number of S 0 which may be adjacent to S in the dependency graph is at most µ 4 4 ¶ (2k d ) (s − 1) ≤ (2k 4 d4 )k , s−1 since s ≤ k + 1. This completes the proof. ¤ Assuming that Claim 5.4 holds, by the second statement of Fact 5.5, we can easily see that the condition (pd < 1/4) of the Local Lemma holds, thus by this lemma P r(∧v∈S DS ) ≥ (1 − 60 × 2k e−5000k

2

(log

1 θ +log d)

≥ exp(−100 × 2k |S|e−5000k (5.8)

2

(log

)|S|

1 θ +log d)

≥ exp(−200 × 2k (k + 1)s−2 mde−5000k

2

(log

1 θ +log d)

),

which implies Claim 5.3. The introduction of important sets is crucial because it reduces the number of sets we need to consider significantly. If instead of important sets, we consider all subsets of size s − 1 (there are Θ(ms−1 ) of them), then in (5.8) we have to write ms−1 instead of m in

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VAN. H VU

the exponent. This would make the argument fall apart since in this case (5.8) cannot be compared with the probabilities in Claim 4.2.1. Now the only thing left is to prove Claim 5.4. Fortunately, this proof is only a slight modification of the proof of Claim 4.2.3 and, with the polynomial method in hand, we do not need any further trick. Let 1 < j < s be an arbitrary integer. Consider a set v of j arbitrary vertices v1 , . . . , vj . We are interested in the random variable Xv = codeg 0 (v) which is the codegree of v1 , . . . , vj in F 0 . Similar to Xv (see Section 4.3), Xv could be expressed as follows Xv =

X Y Y ( (1 − tf )(1 − ρu )). e⊃v u∈e\v f 3u

Again we shall omit the unnecessary subindex v and write X instead of Xv . Following the proof of Claim 4.3, we can bound X from below by X1 and from above by X2 where X1 =

X e⊃v

(1 −

X X ( tf + ρu )) = codeg(v) − U1 (v), u∈e\v f 3u

and X2 = X1 + Y1 + Y2 + Y3 , where

Y1 =

X

X

X

tf tf 0

e⊃v u,u0 ∈e\v f 3u,f 0 3u0

Y2 =

X

X

X

tf ρu0

e⊃v u,u0 ∈e\v f 3u

Y3 =

X

X

ρu ρu 0 .

e⊃v u6=u0 ∈e\v

Notice that here we only need an upper bound for X, thus the proof is in fact a little bit simpler than that of Claim 4.2.3 (where we need both upper and lower bounds). We shall require the following two facts, which are the analogs of Facts 4.3.1 and 4.3.2, respectively. Fact 5.6. With probability at least 1 − 3exp(−θ3 bj /100bj+1 ) the following holds X1 ≤ bj (1 − kpdmax + 4kθ2 ). Fact 5.7. With probability at least 1 − 27exp(− 1018 θ(bj /bj+1 )1/2 ) the following holds X2 − X1 ≤ 10θ2 k 2 bj . Facts 4.3.1, 4.3.2, 5.6 and 5.7 together with assumptions (2) and (3) of Lemma 5.2 imply that the probability that there is some v ∈ S that is not normal or there is a set v ⊂ S whose codegree does not decrease properly is at most

NEARLY PERFECT MATCHINGS IN HYPERGRAPHS

(s − 1) × 30e−5000k ≤ 30 × 2k e−5000k

2

2

(log

(log

1 θ +log d)

1 θ +log d)

+ (2s−1 − (s − 1)) × 30 × e−5000k

31

2

(log

1 θ +log d)

,

which proves Claim 5.4. Proof of Fact 5.6. We have X1 (v) = codeg(v) − U1 (v) where X X X U1 (v) = ( tf + ρu ). e⊃v u∈e\v f 3u

If codeg(v) ≤ 0.9bj then Fact 5.6 holds with probability 1. Thus we can assume that codeg(v) ≥ 0.9bj . Similar to (4.3.2), we can show codeg(v)kdmax p ≥ E(U1 ) ≥ codeg(v)((k − j + 1)dmax p − (k − j + 1)θ2 ). For any vertex u, u can appears in at most bj+1 edges containing v. Thus each ρu is repeated in U1 at most bj+1 times. Similarly, each tf is repeated in U1 at most (k + 1)bj+1 ≤ 2kbj+1 times. Therefore, U1 is a linear function with coefficients at most 2kbj+1 . By Corollary 2.5 P r(U1 ≤ codeg(v)((k − j + 1)dmax p − 4kθ2 ))) ≤ P r(|U1 − E(U1 )| ≥ 3kθ2 codeg(v)) ≤ 3exp(−0.01(3kθ2 codeg(v))2 /E(U1 )2kbj+1 ) ≤ 3exp(−0.02θ3 codeg(v)/bj+1 ) ≤ 3exp(−θ3 bj /100bj+1 ) This completes the proof of Fact 5.6. ¤ The proof of Fact 5.7 can be obtained by modifying the proof of Fact 4.3.2 in the way the proof of Fact 5.6 modifies the proof of Fact 4.3.1. We omit the detail. Algorithmic efficiency. Reasoning as in Remark 4.3.3, one can see that when D is a positive power of n, then the proof can be converted to an efficient (randomized) algorithm. Added in proof. • After this paper was completed, we realized that one can weaken the condition (3) of Theorem 1.2.2 to the following x2 ≤ Dj /Dj+1 , for all j ≤ s − 1, which results in a stronger theorem. • Strengthening Theorem 1.1.1 in another direction, Pippenger and Spencer [PS] shown that the chromatic index of the hypergraph in Theorem 1.1.1 is D(1 + o(1)). The same bound was proved for the list-chromatic index by Kahn [Kah2]. Using the techniques provided in this paper combined with new arguments, we can prove a variant of Theorem 1.2.2 for the chromatic index (and also the list-chromatic index) of a hypegraph, given its codegrees. This gives a more accurate estimate on the error term o(1) in PippengerSpencer’s and Kahn’s result. The proofs of the facts stated above are complicated and will appear elsewhere.

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• We have learnt recently that Kim [Kim] improved the bound in Corollary 1.2.5 by removing the constant 3 in the exponent. His proof deals with Steiner systems directly and does not use the hypergraph setting. Acknowledgement. We would like to thank the referees and E. Vigoda for several comments which lead to a significant improvement on the presentation of the paper. REFERENCES [AKS] N. Alon, J. H. Kim and J. Spencer, Nearly perfect matching in regular simple hypergraphs, Israel J. Math., 100, 171-187 (1997). [AS] N. Alon and J. Spencer, The probabilistic method, Wiley, 1992. [EH] P. Erd˝os and H. Hanani, On a limit theorem in combinatorial analysis, Publ. Math. Debrecen, 10, 10-13 (1963). [EL] P. Erd˝os and L. Lov´asz, Problems and results on 3-chromatic hypergraphs and some related questions, in A. Hajnal et al., eds., Infinite and Finite Sets, North Holland, Amsterdam, 609-628. [FR] P. Frankl and V. R¨odl, Near perfect covering in graphs and hypergraphs, Eur. J. Combin., 6, 317-326 (1985). [Gra] D. Grable, More than nearly perfect packings in partial designs, Combinatorica, to appear. [Gra2] D. Grable, Nearly perfect hypergraph packing in NC, Inform. Process Lett., 60, 295-299 (1996). [GKPS] D. Gordon, G. Kuperberg, O. Patashnik and J. Spencer, Asymptotically optimal covering designs, J. Combin Theory Ser. A, 75, 270-280 (1996). [Kah] J. Kahn, One some hypergraph problems of P. Erd˝os and the asymptotics of matchings, covers and colorings, The mathematics of P. Erd˝ os, edited by R. Graham and J. Nesetril, Springer 1997. [Kah2] J. Kahn, Asymptotically good list-colorings, J. Combinatorial Th. (A), 73, 1–59 (1996). [KR] A. Kostochka and V. R¨odl, Partial Steiner systems and matchings in hypergraphs, Random Struc. and Algorithms, 13, 335-347 (1997). [Kuz] N. Kuzjurin, On the difference between asymptotically good parkings and coverings, Eur. J. Combin., 16, 35-40 (1995). [Kuz2] N. Kuzjurin, Almost optimal explicit constructions of asymptotically good packings and coverings, in Methods of Combinatorial Optimization, Moscow, Computer Center, 1219 (in Russian), 1997. [Kim] J. H. Kim, manuscript. [KV] J. H. Kim and V. H. Vu, On the concentration of multivariate polynomials and applications, to appear in Combinatorica. [KV2] J. H. Kim and V. H. Vu, Small complete arcs in projective planes, submitted. [Pip] N. Pippenger, Unpublished. [R¨od] V. R¨odl, On a packing and covering problem, Eur. J. Combin. 5, 69-78 (1985). [PS] N. Pippenger and J. Spencer, Asymptotic behaviour of the chromatic index of hypergraphs, J. Combi. Theory Series A, 51, 24-42 (1989).

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[Spe] J. Spencer, Asymptotic packing via a branching process, Random Struc. Alg., 7, 167-172 (1995). [Spe2] J. Spencer, The Erd˝os –Hanani conjecture via Talagrand’s inequality, IMA preprint Series 1202 (1994). [Tal] M. Talagrand, Concentration of Measures and Isoperimetric Inequalities in product spaces, Publications Mathematiques de l’I.H.E.S., 81, 73-205 (1996). [Vu1] V. H. Vu, Average smoothness: new concentration results and applications, manuscript. [Vu2] V. H. Vu, On the list chromatic number of locally sparse graphs, manuscript. [Vu3] V. H. Vu, On some degree conditions which guarantee the upper bound of chromatic (choice) number of random graphs, Journal of Graph Theory, 31, 1999, 201-226. [Vu4] V. H. Vu, On a refinement of Waring’s problem, to appear in Duke M. Journal. [Vu5] V. H. Vu, On the concentration of multi-variate polynomials with small expectation, to appear in this journal.