Ramsey properties of random graphs and Ramsey-minimal pairs of graphs

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Ramsey properties of random graphs and Ramsey-minimal pairs of graphs

P. V. Eufrásio

M.S. Student

Y. Kohayakawa

Advisor

Departamento de Ciência da Computação Instituto de Matemática e Estatística

Universidade de São Paulo

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G→(H₁, ...,H_k)

Definition

Given graphsG,H₁,H₂, ...,H_k, we say thatGisRamseyfor

(H₁, ...,H_k)if every edge-coloring ofGwithk colors creates a

monochromatic copy ofH_i in colorifor some 1≤i≤k.

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G→(H₁, ...,H_k)

Definition

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G→(H₁, ...,H_k)

Definition

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G→(H₁, ...,H_k)

Definition

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G→(H₁, ...,H_k)

Definition

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G→(H₁, ...,H_k)

Definition

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G→(H₁, ...,H_k)

Definition

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G→(H₁, ...,H_k)

Definition

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Random Graphs

Definition

Given 0≤p=p(n)≤1, letGn,pdenote the binomial random graphs onnvertices in which every possible edge is present with probabilityp.

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Threshold Functions

Definition

Given a graph propertyA, we say that

(i) Gn,p satisfiesAalmost surely if limn→∞P[Gn,p∈A] =1 and (ii) Gn,p does not satisfyAalmost surely if limn→∞P[Gn,p ∈A] =0. Definition

A functionl(n)is calledthresholdfor a graph propertyAif it satisfies

(i) ifp(n)<<l(n)thenG_n,p(n)does not satisfyAalmost surely and

(ii) ifp(n)>>l(n)thenG_n,p(n)satisfyAalmost surely or vice versa.

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Definition

(i) Gn,psatisfiesAalmost surely if limn→∞P[Gn,p∈A] =1 and

(ii) Gn,p does not satisfyAalmost surely if limn→∞P[Gn,p ∈A] =0. Definition

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Definition

(i) Gn,psatisfiesAalmost surely if limn→∞P[Gn,p∈A] =1 and (ii) Gn,pdoes not satisfyAalmost surely if limn→∞P[Gn,p ∈A] =0.

Definition

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Definition

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Definition

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Definition

(ii) ifp(n)>>l(n)thenG_n,p(n)satisfyAalmost surely

or vice versa.

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Definition

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First Problem

Theorem 1 (Bollobás and Thomason ’87) Every monotone property has a threshold.

Problem

Given graphsH₁,H₂, ...,H_k, find a threshold function for the property G_n,p→(H₁,H₂, ...,H_k).

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First Problem

Theorem 1 (Bollobás and Thomason ’87) Every monotone property has a threshold.

Problem

Given graphsH₁,H₂, ...,H_k, find a threshold function for the property G_n,p →(H₁,H₂, ...,H_k).

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Some Graph Parameters

Given graphsGandH, we define

d₂(G) :=







e_G−1

vG−2 ifv_G ≥3 1/2 ifG∼=K₂ 0 ife_G =0, and set

m₂(G) :=max

J⊆G d₂(J).

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We also define

d₂(G,H) :=

( _e

H

v_H−2+1/m₂(G) ife_G,e_H ≥1

0 otherwise

and set

m₂(G,H) :=max

J⊆H d₂(G,J).

Definition

Ifm₂(G,H) =d₂(G,H)andm₂(G,H)>d₂(G,J),∀J (H,

H isstrictly balancedw.r.td₂(G, .)

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We also define

d₂(G,H) :=

( _e

H

v_H−2+1/m₂(G) ife_G,e_H ≥1

0 otherwise

and set

m₂(G,H) :=max

J⊆H d₂(G,J).

Definition

Ifm₂(G,H) =d₂(G,H)andm₂(G,H)>d₂(G,J),∀J (H,

H isstrictly balancedw.r.td₂(G, .)

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We also define

d₂(G,H) :=

( _e

H

v_H−2+1/m₂(G) ife_G,e_H ≥1

0 otherwise

and set

m₂(G,H) :=max

J⊆H d₂(G,J).

Definition

Ifm₂(G,H) =d₂(G,H)andm₂(G,H)>d₂(G,J),∀J (H, H isstrictly balancedw.r.td₂(G, .)

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Known Thresholds

Theorem 2 (Rödl and Ruci ´nski, ’95)

Letk ≥2 andHbe a graph that is not a forest. Then there exist constantsc,C≥0 such that

n→∞lim P[G_n,p→(H)_k] =

(0 ifp =p(n)≤cn^−1/m²^(H) 1 ifp =p(n)≥Cn^−1/m²^(H).

Theorem 3 (Kohayakawa and Kreuter, ’97)

Letk ≥2 and 3≤l₁≤l₂· · · ≤l_k be integers. Then there exist constantsc,C≥0 such that

n→∞lim P[Gn,p→(C_l₁, . . . ,C_l_k)] =

(0 ifp=p(n)≤cn^−1/m²^(C^l²^,C^l¹⁾ 1 ifp=p(n)≥Cn^−1/m²^(C^l²^,C^l¹⁾, whereC_l denotes the cycle of lengthl.

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Theorem 2 (Rödl and Ruci ´nski, ’95)

Letk ≥2 andHbe a graph that is not a forest. Then there exist constantsc,C≥0 such that

n→∞lim P[G_n,p→(H)_k] =

(0 ifp =p(n)≤cn^−1/m²^(H) 1 ifp =p(n)≥Cn^−1/m²^(H). Theorem 3 (Kohayakawa and Kreuter, ’97)

Letk ≥2 and 3≤l₁≤l₂· · · ≤l_k be integers. Then there exist constantsc,C≥0 such that

n→∞lim P[Gn,p→(C_l₁, . . . ,C_l_k)] =

(0 ifp=p(n)≤cn^−1/m²^(C^l²^,C^l¹⁾ 1 ifp=p(n)≥Cn^−1/m²^(C^l²^,C^l¹⁾,

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Conjecture 4 (Kohayakawa and Kreuter, ’97)

LetGandH be graphs that are not forests and satisfy

m₂(G)≤m₂(H). Then there exist constantsc,C>0 such that

n→∞lim P[Gn,p→(G,H)] =

(0 ifp=p(n)≤cn^−1/m²^(G,H) 1 ifp=p(n)≥Cn^−1/m²^(G,H).

Theorem 5 (Kohayakawa, Schacht and Spöhel, ’2011)

LetGandH be graphs that are not matchings, withH strictly balanced w.r.t. d₂(G, .). Then there exists a constantC>0 such that for

p =p(n)≥Cn^−1/m²^(G,H)we have

n→∞lim P[G_n,p→(G,H)] =1.

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Conjecture 4 (Kohayakawa and Kreuter, ’97)

LetGandH be graphs that are not forests and satisfy

m₂(G)≤m₂(H). Then there exist constantsc,C>0 such that

n→∞lim P[Gn,p→(G,H)] =

(0 ifp=p(n)≤cn^−1/m²^(G,H) 1 ifp=p(n)≥Cn^−1/m²^(G,H). Theorem 5 (Kohayakawa, Schacht and Spöhel, ’2011)

LetGandH be graphs that are not matchings, withH strictly balanced w.r.t. d₂(G, .). Then there exists a constantC>0 such that for

p=p(n)≥Cn^−1/m²^(G,H)we have

lim P[G →(G,H)] =1.

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Together with the results for complete graphs in [1], Theorem 5 implies the following corollary.

Corollary 6

Let 3≤l<r be integers. Then there exist constantsc,C >0 such that

n→∞lim P[G_n,p→(K_l,K_r)] =

(0 ifp=p(n)≤cn^−1/m²^(K^l^,K^r⁾ 1 ifp=p(n)≥Cn^−1/m²^(K^l^,K^r⁾, whereK_l is the complete graph onlvertices.

[1]M. Marciniszyn, J. Skokan, R. Spöhel, A. Steger.Asymmetric Ramsey properties of random

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Open Problems

Problem 1

Prove the 0-statement of the Conjecture 4.

Problem 2

Prove the 1-statement of the Conjecture 4 without using the balancedness assumption.

Problem 3

Find a threshold function for the general Ramsey property G_n,p→(H₁,H₂, ...,H_k).

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Open Problems

Problem 1

Problem 2

Problem 3

Find a threshold function for the general Ramsey property G_n,p→(H₁,H₂, ...,H_k).

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Open Problems

Problem 1

Problem 2

Problem 3

Find a threshold function for the general Ramsey property G_n,p →(H₁,H₂, ...,H_k).

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Second Problem

Definition

LetC(G,H)be the the class of all graphsF that are minimal with respect to the propertyF →(G,H).

Definition

A pair of graphs (G,H) is

Ramsey-finiteifC(G,H)is finite.

Ramsey-infiniteifC(G,H)is infinite.

highly Ramsey-infiniteif there are at least 2^cn² non-isomorphic graphs on at mostnvertices inC(G,H).

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Second Problem

Definition

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Second Problem

Definition

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Second Problem

Definition

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Second Problem

Problem

Given graphsGandH,

is the pair(G,H)Ramsey-finite or Ramsey-infinite?

if the pair(G,H)is Ramsey-infinite, is it highly Ramsey-infinite?

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Second Problem

Problem

Given graphsGandH,

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Second Problem

Problem

Given graphsGandH,

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Ramsey-finite Pairs

The pair(G,H)is Ramsey-finite if

GorHis a matching^[1]or GandHare both odd stars^[2].

[1]S. Burr, P. Erd ˝os, R. Faudree, C. Rousseau, R. Schelp.A class of Ramsey-finite graphs.Proc. of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing. Utilitas Math., (1978) 171-180.

[2]______Ramsey-minimal graphs for forests.Discrete Math. 38 (1982), no. 1, 23-32.

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The pair(G,H)is Ramsey-finite if GorHis a matching^[1]or

GandHare both odd stars^[2].

[1]S. Burr, P. Erd ˝os, R. Faudree, C. Rousseau, R. Schelp.A class of Ramsey-finite graphs.Proc.

of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing. Utilitas Math., (1978) 171-180.

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The pair(G,H)is Ramsey-finite if GorHis a matching^[1]or GandHare both odd stars^[2].

[1]S. Burr, P. Erd ˝os, R. Faudree, C. Rousseau, R. Schelp.A class of Ramsey-finite graphs.Proc.

of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing. Utilitas Math., (1978) 171-180.

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Ramsey-infinite Pairs

The pair(G,H)is Ramsey-infinite if

GorHis a forest (and is not one of the finite cases)^[1], GandHare both non-bipartite^[2],

GandHare both 3-conected^[2]or

Gis a cycle of lengthlandH is a graph containing no induced cycle of lengthl or more^[3].

[1]S. Burr, P. Erd ˝os, R. Faudree, C. Rousseau, R. Schelp, T. Luczak. Over several papers. (1980, 1981, 1991, 1994)

[2]J. Nesetril, V. Rödl.Partition of vertices.Comment. Math. Univ. Carolinae 17 (1976), no. 1, 85-95.

[3]B. Bollobás, J. Donadelli, Y. Kohayakawa, R. H. Schelp.Ramsey minimal graphs.J. of the Brazilian Computer Society 7 (2002), no. 3, 27-37.

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GorHis a forest (and is not one of the finite cases)^[1],

GandHare both non-bipartite^[2], GandHare both 3-conected^[2]or

[1]S. Burr, P. Erd ˝os, R. Faudree, C. Rousseau, R. Schelp, T. Luczak. Over several papers.

(1980, 1981, 1991, 1994)

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(1980, 1981, 1991, 1994)

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(1980, 1981, 1991, 1994)

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(1980, 1981, 1991, 1994)

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Highly Ramsey-infinite Pairs

The pair(G,H)is highly Ramsey-infinite if

GandHare both cliques^[1]or G=H is an odd cycle^[2,3].

[1]V. Rödl, M. Siggers.On Ramsey minimal graphs.SIAM J Discrete Math 22 (2008), no. 2, 467-488.

[2]M. Siggers.On highly Ramsey-infinite graphs.J. Graph Theory Vol. 59 (2008), no. 2, 97-114. [3]M. Siggers.Five cycles are highly Ramsey-infinite.To appear in Kyungpook Mathematical Journal, 2012.

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The pair(G,H)is highly Ramsey-infinite if GandHare both cliques^[1]or

G=H is an odd cycle^[2,3].

[2]M. Siggers.On highly Ramsey-infinite graphs.J. Graph Theory Vol. 59 (2008), no. 2, 97-114. [3]M. Siggers.Five cycles are highly Ramsey-infinite.To appear in Kyungpook Mathematical Journal, 2012.

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The pair(G,H)is highly Ramsey-infinite if GandHare both cliques^[1]or

G=H is an odd cycle^[2,3].

[2]M. Siggers.On highly Ramsey-infinite graphs.J. Graph Theory Vol. 59 (2008), no. 2, 97-114.

[3]M. Siggers.Five cycles are highly Ramsey-infinite.To appear in Kyungpook Mathematical Journal, 2012.

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Open Problems

The problem of deciding whether(G,H)is Ramsey-finite or infinite remains open for bipartite graphs with vertex connectivity 2.

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Discussion of Theorem 5 For any graphHwithv_H ≥3, let

m^∗(H) := min

J⊂H:2≤v_J<vH

e_H−e_J v_H−v_J and, ifH is nonempty,

x^∗(H) := m^∗(H) e_H−m^∗(H)(v_H−2) Definition

For anyρ >0 and 0<d ≤1, a graphF onnvertices is said to be (ρ,d)-dense if for every subsetS⊆V(G)with|S| ≥ρnwe have

e(F[S])≥d |S|

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Discussion of Theorem 5

Lemma 6 - Simplified version

LetHbe a nonempty graph withv_H ≥3. For any 0<d ≤1, there exists a positive constantρsuch that ifp≤n^−1/m^∗^(H) the following holds: ifF ⊆K_nis a(ρ,d)-dense graph, then, with "high" probability, the graphF∩G_n,pcontains "many" edge-disjoint copies ofH.

LetHbe a nonempty graph withv_H ≥3. For any nonempty graphG satisfyingm₂(G)<x^∗(H)and

Cn^−1/m²^(G,H)≤p≤n^−1/m^∗^(H),

with "high" probability, every red-blue edge-coloring ofGn,pthat does not contain a blue copy ofHcontains "many" red copies ofG.

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LetHbe a nonempty graph withv_H ≥3. For any 0<d ≤1, there exists a positive constantρsuch that ifp≤n^−1/m^∗^(H) the following holds: ifF ⊆K_nis a(ρ,d)-dense graph, then, with "high" probability, the graphF∩G_n,pcontains "many" edge-disjoint copies ofH.

LetHbe a nonempty graph withv_H ≥3. For any nonempty graphG satisfyingm₂(G)<x^∗(H)and

Cn^−1/m²^(G,H)≤p≤n^−1/m^∗^(H),

with "high" probability, every red-blue edge-coloring ofG_n,pthat does

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Lemma 6

LetHbe a nonempty graph withv_H ≥3. For any 0<d ≤1, there exist positive constantsρ,n₀andbsuch that forn≥n₀andp≤n^−1/m^∗^(H) the following holds: ifF ⊆K_nis a(ρ,d)-dense graph onnvertices, then, with probability at least 1−2^−bn^vH^p^eH⁺¹, the graphF ∩Gn,p

contains a family of at leastbn^v^Hp^e^H pairwise edge-disjoint copies ofH.

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Lemma 7

LetHbe a nonempty graph withv_H ≥3. For any nonempty graphG satisfyingm₂(G)<x^∗(H)there exist positive constantsa,b,Candn₀ such that forn≥n₀and

Cn^−1/m²^(G,H)≤p≤n^−1/m^∗^(H),

with probability at least 1−2^−bn^vH^pêH, every red-blue-coloring of E_H(G_n,p)that does not contain a blue copy ofHcontains at least an^v^G(n^v^H⁻²pê^H)ê^G manyH-covered red copies ofG.

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