Monochromatic Schur Triples in Randomly Perturbed Dense Sets of Integers

Aigner-Horev, Elad; Person, Yury

doi:10.1137/18m1227007

Cited by 9 publications

(17 citation statements)

References 31 publications

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“…[5,21,28,33]). Further, very recently an analogous model of randomly perturbed sets of integers has been studied [1].…”

Section: 2mentioning

confidence: 99%

“…Given a graph H and a constant 0 < ε < 1/2, there are positive constants q = q(ε, H), β = β(ε, H) and κ = κ(ε, H) such that for every n ∈ N, there is a function ψ : P(E(K n )) q → P(E(K n )) with the following properties. Let T := {(T (1) , . .…”

Section: By Lemma 22(i) With = 1 the Number Of Vertices Inmentioning

confidence: 99%

“…We now apply Corollary A.4 with H := H 2 and ε := π/2. This gives us constants q, β and κ, together with a collection T := {(T (1) , . .…”

Section: By Lemma 22(i) With = 1 the Number Of Vertices Inmentioning

confidence: 99%

“…Then there exists a constant c = c(r) and a function ψ :P([N ]) q → P([N ]), where q ≤ c log(1/ε), with the following properties. Let T := {(T(1) , . .…”

mentioning

confidence: 99%

“…, T (q) ) ∈ P([N ]) q : T (j) ≤ cτ N, j ∈ [q]}, and let C := {ψ(T ) : T ∈ T }. Then (a) for every set I ⊂ [N ] for which e(Γ[I]) ≤ 24εr!rτ r e(Γ), there exists T = (T(1) , . .…”

mentioning

confidence: 99%

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Ramsey properties of randomly perturbed graphs: cliques and cycles

Das

Treglown

2020

Combinator. Probab. Comp.

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Given graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, K t ) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, K t ). We make significant progress on this question, giving a precise solution in the case when H1 = K s and H2 = K t where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, K t ) -Ramsey question. Moreover, we give bounds for the corresponding (K4, K t ) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem. We also give a precise solution to the analogous question in the case when both H1 = C s and H2 = C t are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (C s , K t ) -Ramsey (for odd s ≽ 5 and t ≽ 4). To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

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“…[5,21,28,33]). Further, very recently an analogous model of randomly perturbed sets of integers has been studied [1].…”

Section: 2mentioning

confidence: 99%

Section: By Lemma 22(i) With = 1 the Number Of Vertices Inmentioning

confidence: 99%

“…We now apply Corollary A.4 with H := H 2 and ε := π/2. This gives us constants q, β and κ, together with a collection T := {(T (1) , . .…”

Section: By Lemma 22(i) With = 1 the Number Of Vertices Inmentioning

confidence: 99%

“…Then there exists a constant c = c(r) and a function ψ :P([N ]) q → P([N ]), where q ≤ c log(1/ε), with the following properties. Let T := {(T(1) , . .…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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