The Ramsey number of the clique and the hypercube - Illinois
The Ramsey number of the clique and the hypercube Jozef Skokan £ondon $chool of Cconomics [email protected] Gonzalo Fiz Pontiveros, Simon Griffiths, Rob Morris, David Saxton (IMPA) September 3, 2013
Ramsey theory, basics
B r(n, s): smallest N such that any colouring of the edges of KN with blue and red contains a blue clique on n vertices or a red clique on s vertices
B r(3, 3) = 6 B Ramsey, 1930: r(n, s) < ∞ B Celebrated problem: determining/estimating r(n, n) ˝ and Szekeres, ´ 1935: r(n, n) ≤ 22n B Erdos ˝ 1947: 2n/2 ≤ r(n, n) B Erdos, T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
1
Graph Ramsey theory
B r(G, H): smallest N such that any colouring of the edges of KN with blue and red contains a blue G or a red H B Existence: trivial, from r(G, H) ≤ r(|G|, |H|) < ∞ Motivation: test the limits of current methods and develop a new ones that could be (possibly) used to improve bounds on r(n, n) Examples: ´ Gyarf ´ as, ´ 1967: r(Pn, Pm) = n + bm/2c − 1 for n ≥ m ≥ 2 B Gerencser, ´ B Chvatal, 1977: r(Tn, Km) = (m − 1)(n − 1) + 1 for m ≥ 2, any tree Tn T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
2
Simple Lower Bound ´ Theorem 1 (Chvatal, Harary, 1972). If G is connected, then r(G, H) ≥ (χ(H) − 1)(|G| − 1) + 1. Proof.
B take χ(H) − 1 disjoint BLUE cliques of size |G| − 1; B colour by RED all pairs between two red cliques; B G does not fit to one BLUE clique of size |G| − 1; B RED subgraph has chromatic number χ(H) − 1 and cannot contain H.
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
3
Simple Lower Bound, cont.
B σ(H): the minimum size of a colour class taken over all χ(H)-colourings of H B σ(Km) = 1, σ(Pm) = bm/2c, σ(C2m−1) = 1, σ(C2m) = m
Theorem 2 (Burr, 1980). If G is connected and |G| ≥ σ(H), then r(G, H) ≥ (χ(H) − 1)(|G| − 1) + σ(H).
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
4
Ramsey Goodness
B G is H-good if r(G, H) = (χ(H) − 1)(|G| − 1) + σ(H) Examples: ´ B Chvatal, 1977: all trees are Km-good for m ≥ 2 ˝ 1973: Cn is Km-good for n ≥ m2 − 2 B Bondy, Erdos, B Nikiforov, 2008: Cn is Km-good for n ≥ 4m + 2 ˝ B Conjecture (Erdos): Cn is Km-good for n ≥ m B Interesting problem: behaviour of r(Km, Cn) for fixed m ≥ 3 T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
5
˝ conjectures Burr–Erdos
6
˝ Which graph parameters control the growth of r(G, H) ? Burr, Erdos: Conjecture 1 (Burr, 1981). Fix a graph H and ∆ ∈ N. connected graph G with ∆(G) ≤ ∆ is H-good:
Then any large
r(G, H) = (χ(H) − 1)(|G| − 1) + σ(H).
G is ∆-degenerate: every subgraph of G has a vertex of degree ≤ ∆ ˝ 1983). Fix m ≥ 3 and ∆ ≥ 1. Then every large Conjecture 2 (Burr and Erdos, connected ∆-degenerate graph is Km-good. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
˝ conjectures, cont. Burr–Erdos
˝ B Burr, Erdos: gave a list of test graphs that should be Km-good: wheels K1 + Cn, certain subdivisions of Kn, K1 + Cnk , . . . , hypercubes.
B Nikiforov and Rousseau, 2009: resolved (positively) all but one of these questions.
B remaining question (modest version): Is Qn K3-good for large n?
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
7
Conjectures 1 and 2 are false
B Brandt, 1996: there exists a 168-regular graph that is not K3-good B Nikiforov and Rousseau, 2009: almost all 100-regular graphs are not K3good B counterexamples have good expansion properties B what if we limit expansion? B bandwidth bw(G): smallest k such that V (G) has ordering v1, v2, . . . , v|G|: if vivj ∈ E(G) then |i − j| ≤ k. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
8
Conjecture 1 ‘rescued’
Theorem 3 (Allen, Brightwell, S., 2013). Fix k ∈ N and a graph H. Then any sufficiently large connected graph G with bw(G) ≤ k is H-good: r(G, H) = (χ(H) − 1)(|G| − 1) + σ(H).
Note: k and H can grow with |G| ‘reasonably fast’ Theorem 4 (Allen, Brightwell, S., 2013). Fix ∆ ∈ N and a graph H. Then any sufficiently large connected graph G with ∆(G) ≤ ∆ and bw(G) = o(|G|) is H-good. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
9
Hypercube Qn
10
V (Qn): all sequences x = x1x2 . . . xn with entries from {0, 1} E(Qn): xy is an edge if x and y differ in exactly one position 110 x 100 x
111
x
011
x
101
010 x 000 x
x
x
001 hypercube Q3
√ Properties: n-regular on 2 vertices, bipartite, bw(Qn) ∼ 2 / n n
n
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Are hypercubes good?
We would like to see whether, for H fixed and n large, r(Qn, H) = (χ(H) − 1)(2n − 1) + σ(H). For H = K3, is r(Qn, K3) = 2n+1 − 1?
Theorems 3 and 4 do not apply: bw(Qn) and ∆(Qn) grow too fast.
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
11
2n+1 − 1 ≤ R(Qn, K3) ≤ (n + 1)(2n − 1) + 1 In any 2-colouring of KN with N = (n + 1)(2n − 1) + 1 and no red K3 B the red neighbours of any vertex form a blue clique B hence the red degree of any vertex ≤ 2n − 1 (or blue Qn) B take any ordering v1, . . . , v2n of V (Qn), embed greedily B vi has at most n neighbours in v1, . . . , vi−1 and they have ≤ n(2n − 1) red neighbours in KN B vi has ≥ N − n(2n − 1) − (i − 1) > 0 available images We only used that Qn is n-regular. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
12
Results
13
B Conlon, Fox, Lee, Sudakov (2012+) R(Qn, K3) ≤ 7200 · 2n for n ≥ 6, and R(Qn, H) ≤ cH 2n for H fixed and n large. B Fiz Pontiveros, Griffiths, Morris, Saxton, S. (2013+) R(Qn, K3) = (1 + o(1))2n+1as n → ∞. B Fiz Pontiveros, Griffiths, Morris, Saxton, S. (2013+) Qn is H-good for any fixed H and n large. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Some ideas from R(Qn, K3) ≤ 7000 · 2n initial subcube of co-dimension d: for x = (x1, . . . , xd) ∈ {0, 1}d, let n Qx = y = (y1, y2, . . . , yn) ∈ {0, 1} : yi = xi for each 1 ≤ i ≤ d Note: every vertex v ∈ V (Qx) has exactly d neighbours in V (Qn) \ V (Qx) d
d0
Observations: for x ∈ {0, 1} and z ∈ {0, 1} , v ∈ Qx, and w ∈ Qz, B if Qx ∩ Qz 6= ∅, then Qx ⊆ Qz or Qz ⊆ Qx. B Qx ∩Qz = ∅ iff x and z differ in at least one of the first min{d, d0} coordinates. B when Qx ∩ Qz = ∅, vw ∈ E(Qn) iff x and z differ in precisely one of the first min{d, d0} coordinates. We call such Qx and Qz adjacent. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
14
15
Find B decomposition of Qn = Q1 ∪ Q2 ∪ · · · ∪ Q` into initial subcubes, Qi of codimension di, d1 ≤ d2 ≤ . . . d`, B a collection of BLUE cliques T 1, T 2 . . . , T ` in KN , such that B |T i| = 2|Qi| = 2 · 2n−di , B if Qi and Qj are adjacent, i < j, then each vertex of T j has at most |T i|/4di RED neighbours in Ti. Embed Q` → T `, Q`−1 → T `−1 . . . Q1 → T 1 sequentially and greedily. Constant 7200 comes from the construction of this setup. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Some ideas from R(Qn, K3) = (1 + o(1))2n+1
Take 2-coloured KN with N = (1 + 2)2n+1 with no RED K3. Case 1: there exists C, |C| > (1 + )2n such that every vertex v ∈ C has at most 2n/ log log log n RED neighbours in C. Then we can find a setup similar to the previous proof and embed Qn to C. Case 2: there exist a collection of (m, s)-snakes covering > (1 + )2n with small RED degrees between them (m, s)-snake: a collection of BLUE cliques of size m connected by BLUE Ks,s
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
16
(m, s)-snakes contain large powers of the path
Motivation: any ` vertex graph with bandwidth s is a subgraph of P`s, we can embed P`s to an (m, s)-snake S if |S| ≥ `. In particular, Qn ⊂ P`s, where ` ≥ 2n and s ≥ 2n/n1/3 > bw(Qn). T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
17
Precise statement
18
Fix n large, let G be a two-coloured complete graph with no blue triangles, and with 2n ≤ v(G) ≤ 2n+2. Then there exists a partition of V (G) into sets C ∪ S1 ∪ · · · ∪ Sr , for some r ≥ 0, such that the following conditions hold: (a) e GB [C] ≤
2n |C| log log n
and, for every i ∈ [r], there exists n−1/3 ≤ si · 2−n ≤ n−1/4 such that (b) GR[Si] contains a spanning (m, si)-snake Si, where m = (c) |NB (v) ∩ Si| ≤
si log log n
2n−1 log log n .
for every v ∈ Si+1 ∪ · · · ∪ Sr . T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Questions ....
19
B What additional conditions imply that ∆-degenerate graph G is H-good ? B Is it true that r(Qn, Qn) = O(2n) ? (Fox, Sudakov 2009: ≤ n22n+5) B Is it true that r(Qn, Qm) = O(2n) for m growing with n ? B For large G with ∆(G) ≤ ∆ and bw(G) = o(|G|), it appears that r(G, G) ≤ (χ(G) + α)|G| + β. What are α and β? T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
B Is it true that for all graphs G with maximum degree at most ∆, we have r(G, G) ≤ 2c∆|G|? Conlon, Fox, Sudakov 2009: ≤ 2c∆ log ∆|G| B Is it true that for all ∆-degenerate graphs G, we have r(G, G) ≤ c(∆)|G|? Kostochka, Sudakov 2003:≤ |G|1+o(1)
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
20
Ramsey theory, basics
B r(n, s): smallest N such that any colouring of the edges of KN with blue and red contains a blue clique on n vertices or a red clique on s vertices
B r(3, 3) = 6 B Ramsey, 1930: r(n, s) < ∞ B Celebrated problem: determining/estimating r(n, n) ˝ and Szekeres, ´ 1935: r(n, n) ≤ 22n B Erdos ˝ 1947: 2n/2 ≤ r(n, n) B Erdos, T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
1
Graph Ramsey theory
B r(G, H): smallest N such that any colouring of the edges of KN with blue and red contains a blue G or a red H B Existence: trivial, from r(G, H) ≤ r(|G|, |H|) < ∞ Motivation: test the limits of current methods and develop a new ones that could be (possibly) used to improve bounds on r(n, n) Examples: ´ Gyarf ´ as, ´ 1967: r(Pn, Pm) = n + bm/2c − 1 for n ≥ m ≥ 2 B Gerencser, ´ B Chvatal, 1977: r(Tn, Km) = (m − 1)(n − 1) + 1 for m ≥ 2, any tree Tn T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
2
Simple Lower Bound ´ Theorem 1 (Chvatal, Harary, 1972). If G is connected, then r(G, H) ≥ (χ(H) − 1)(|G| − 1) + 1. Proof.
B take χ(H) − 1 disjoint BLUE cliques of size |G| − 1; B colour by RED all pairs between two red cliques; B G does not fit to one BLUE clique of size |G| − 1; B RED subgraph has chromatic number χ(H) − 1 and cannot contain H.
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
3
Simple Lower Bound, cont.
B σ(H): the minimum size of a colour class taken over all χ(H)-colourings of H B σ(Km) = 1, σ(Pm) = bm/2c, σ(C2m−1) = 1, σ(C2m) = m
Theorem 2 (Burr, 1980). If G is connected and |G| ≥ σ(H), then r(G, H) ≥ (χ(H) − 1)(|G| − 1) + σ(H).
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
4
Ramsey Goodness
B G is H-good if r(G, H) = (χ(H) − 1)(|G| − 1) + σ(H) Examples: ´ B Chvatal, 1977: all trees are Km-good for m ≥ 2 ˝ 1973: Cn is Km-good for n ≥ m2 − 2 B Bondy, Erdos, B Nikiforov, 2008: Cn is Km-good for n ≥ 4m + 2 ˝ B Conjecture (Erdos): Cn is Km-good for n ≥ m B Interesting problem: behaviour of r(Km, Cn) for fixed m ≥ 3 T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
5
˝ conjectures Burr–Erdos
6
˝ Which graph parameters control the growth of r(G, H) ? Burr, Erdos: Conjecture 1 (Burr, 1981). Fix a graph H and ∆ ∈ N. connected graph G with ∆(G) ≤ ∆ is H-good:
Then any large
r(G, H) = (χ(H) − 1)(|G| − 1) + σ(H).
G is ∆-degenerate: every subgraph of G has a vertex of degree ≤ ∆ ˝ 1983). Fix m ≥ 3 and ∆ ≥ 1. Then every large Conjecture 2 (Burr and Erdos, connected ∆-degenerate graph is Km-good. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
˝ conjectures, cont. Burr–Erdos
˝ B Burr, Erdos: gave a list of test graphs that should be Km-good: wheels K1 + Cn, certain subdivisions of Kn, K1 + Cnk , . . . , hypercubes.
B Nikiforov and Rousseau, 2009: resolved (positively) all but one of these questions.
B remaining question (modest version): Is Qn K3-good for large n?
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
7
Conjectures 1 and 2 are false
B Brandt, 1996: there exists a 168-regular graph that is not K3-good B Nikiforov and Rousseau, 2009: almost all 100-regular graphs are not K3good B counterexamples have good expansion properties B what if we limit expansion? B bandwidth bw(G): smallest k such that V (G) has ordering v1, v2, . . . , v|G|: if vivj ∈ E(G) then |i − j| ≤ k. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
8
Conjecture 1 ‘rescued’
Theorem 3 (Allen, Brightwell, S., 2013). Fix k ∈ N and a graph H. Then any sufficiently large connected graph G with bw(G) ≤ k is H-good: r(G, H) = (χ(H) − 1)(|G| − 1) + σ(H).
Note: k and H can grow with |G| ‘reasonably fast’ Theorem 4 (Allen, Brightwell, S., 2013). Fix ∆ ∈ N and a graph H. Then any sufficiently large connected graph G with ∆(G) ≤ ∆ and bw(G) = o(|G|) is H-good. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
9
Hypercube Qn
10
V (Qn): all sequences x = x1x2 . . . xn with entries from {0, 1} E(Qn): xy is an edge if x and y differ in exactly one position 110 x 100 x
111
x
011
x
101
010 x 000 x
x
x
001 hypercube Q3
√ Properties: n-regular on 2 vertices, bipartite, bw(Qn) ∼ 2 / n n
n
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Are hypercubes good?
We would like to see whether, for H fixed and n large, r(Qn, H) = (χ(H) − 1)(2n − 1) + σ(H). For H = K3, is r(Qn, K3) = 2n+1 − 1?
Theorems 3 and 4 do not apply: bw(Qn) and ∆(Qn) grow too fast.
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
11
2n+1 − 1 ≤ R(Qn, K3) ≤ (n + 1)(2n − 1) + 1 In any 2-colouring of KN with N = (n + 1)(2n − 1) + 1 and no red K3 B the red neighbours of any vertex form a blue clique B hence the red degree of any vertex ≤ 2n − 1 (or blue Qn) B take any ordering v1, . . . , v2n of V (Qn), embed greedily B vi has at most n neighbours in v1, . . . , vi−1 and they have ≤ n(2n − 1) red neighbours in KN B vi has ≥ N − n(2n − 1) − (i − 1) > 0 available images We only used that Qn is n-regular. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
12
Results
13
B Conlon, Fox, Lee, Sudakov (2012+) R(Qn, K3) ≤ 7200 · 2n for n ≥ 6, and R(Qn, H) ≤ cH 2n for H fixed and n large. B Fiz Pontiveros, Griffiths, Morris, Saxton, S. (2013+) R(Qn, K3) = (1 + o(1))2n+1as n → ∞. B Fiz Pontiveros, Griffiths, Morris, Saxton, S. (2013+) Qn is H-good for any fixed H and n large. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Some ideas from R(Qn, K3) ≤ 7000 · 2n initial subcube of co-dimension d: for x = (x1, . . . , xd) ∈ {0, 1}d, let n Qx = y = (y1, y2, . . . , yn) ∈ {0, 1} : yi = xi for each 1 ≤ i ≤ d Note: every vertex v ∈ V (Qx) has exactly d neighbours in V (Qn) \ V (Qx) d
d0
Observations: for x ∈ {0, 1} and z ∈ {0, 1} , v ∈ Qx, and w ∈ Qz, B if Qx ∩ Qz 6= ∅, then Qx ⊆ Qz or Qz ⊆ Qx. B Qx ∩Qz = ∅ iff x and z differ in at least one of the first min{d, d0} coordinates. B when Qx ∩ Qz = ∅, vw ∈ E(Qn) iff x and z differ in precisely one of the first min{d, d0} coordinates. We call such Qx and Qz adjacent. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
14
15
Find B decomposition of Qn = Q1 ∪ Q2 ∪ · · · ∪ Q` into initial subcubes, Qi of codimension di, d1 ≤ d2 ≤ . . . d`, B a collection of BLUE cliques T 1, T 2 . . . , T ` in KN , such that B |T i| = 2|Qi| = 2 · 2n−di , B if Qi and Qj are adjacent, i < j, then each vertex of T j has at most |T i|/4di RED neighbours in Ti. Embed Q` → T `, Q`−1 → T `−1 . . . Q1 → T 1 sequentially and greedily. Constant 7200 comes from the construction of this setup. T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Some ideas from R(Qn, K3) = (1 + o(1))2n+1
Take 2-coloured KN with N = (1 + 2)2n+1 with no RED K3. Case 1: there exists C, |C| > (1 + )2n such that every vertex v ∈ C has at most 2n/ log log log n RED neighbours in C. Then we can find a setup similar to the previous proof and embed Qn to C. Case 2: there exist a collection of (m, s)-snakes covering > (1 + )2n with small RED degrees between them (m, s)-snake: a collection of BLUE cliques of size m connected by BLUE Ks,s
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
16
(m, s)-snakes contain large powers of the path
Motivation: any ` vertex graph with bandwidth s is a subgraph of P`s, we can embed P`s to an (m, s)-snake S if |S| ≥ `. In particular, Qn ⊂ P`s, where ` ≥ 2n and s ≥ 2n/n1/3 > bw(Qn). T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
17
Precise statement
18
Fix n large, let G be a two-coloured complete graph with no blue triangles, and with 2n ≤ v(G) ≤ 2n+2. Then there exists a partition of V (G) into sets C ∪ S1 ∪ · · · ∪ Sr , for some r ≥ 0, such that the following conditions hold: (a) e GB [C] ≤
2n |C| log log n
and, for every i ∈ [r], there exists n−1/3 ≤ si · 2−n ≤ n−1/4 such that (b) GR[Si] contains a spanning (m, si)-snake Si, where m = (c) |NB (v) ∩ Si| ≤
si log log n
2n−1 log log n .
for every v ∈ Si+1 ∪ · · · ∪ Sr . T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
Questions ....
19
B What additional conditions imply that ∆-degenerate graph G is H-good ? B Is it true that r(Qn, Qn) = O(2n) ? (Fox, Sudakov 2009: ≤ n22n+5) B Is it true that r(Qn, Qm) = O(2n) for m growing with n ? B For large G with ∆(G) ≤ ∆ and bw(G) = o(|G|), it appears that r(G, G) ≤ (χ(G) + α)|G| + β. What are α and β? T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
B Is it true that for all graphs G with maximum degree at most ∆, we have r(G, G) ≤ 2c∆|G|? Conlon, Fox, Sudakov 2009: ≤ 2c∆ log ∆|G| B Is it true that for all ∆-degenerate graphs G, we have r(G, G) ≤ c(∆)|G|? Kostochka, Sudakov 2003:≤ |G|1+o(1)
T HE R AMSEY
NUMBER OF THE CLIQUE AND THE HYPERCUBE
20
