Asymmetric Ramsey properties of random graphs involving cliques and cycles

Liebenau, Anita; Mattos, Letícia; Mendonça, Walner; Skokan, Jozef

doi:10.1002/rsa.21106

Cited by 2 publications

(3 citation statements)

References 18 publications

(60 reference statements)

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“…We then apply a new procedure Extend which attaches either a k-graph L ∈ L G ′ or R ∈ R G ′ that contains e to F i . As in Case 1, because the condition in line 6 of Asym-Edge-Col-Bε fails, G ′ ∈ C. 22 We now show that the number of edges of F i increases by at least one in each iteration of the while-loop and that Grow-Alt operates as desired with a result analogous to Claim 7.1.…”

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confidence: 56%

“…This is achieved in the paper of Marciniszyn, Skokan, Spöhel and Steger [23] which deals with the case when H 1 and H 2 are both cliques (and not both K 3 ), albeit with different methods. In [18] and [22], which deal with the cases when H 1 and H 2 are both cycles and H 1 and H 2 are a clique and a cycle, respectively, the authors of both papers show implicitly that B(H 1 , H 2 ) = ∅. Let us also remark that in [23], it is implicitly shown that B(H 1 , H 2 ) with each H i a clique, is non-empty only when H 2 = K 3 or H 1 = H 2 = K 4 , and moreover, | B(H 1 , H 2 )| ≤ 3 in each case.…”

Section: Discussionmentioning

confidence: 99%

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A resolution of the Kohayakawa--Kreuter conjecture for the majority of cases

Bowtell

Hancock

Hyde

2023

Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications

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For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that $G(n,p) \to (H_1,\dots,H_r)$, thereby resolving the $1$-statement of the Kohayakawa--Kreuter conjecture. %We show that to prove the $0$-statement it suffices to prove a deterministic colouring result, which says that if $G$ is not too dense then $G \not \to (H_1,\dots,H_r)$. We extend upon the many partial results for the $0$-statement, by resolving it for a large number of cases, which in particular includes (but is not limited to) when $r \geq 3$, when $H_2$ is strictly $2$-balanced and not bipartite, or when $H_1$ and $H_2$ have the same $2$-densities.

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confidence: 56%

Section: Discussionmentioning

confidence: 99%

A resolution of the Kohayakawa--Kreuter conjecture for the majority of cases

Bowtell

Hancock

Hyde

2023

Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications

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“…In particular, when H is a tree the threshold is p = 1∕N. A corresponding result for hypergraphs was obtained by Friedgut, Rödl, and Schacht [13] and independently by Conlon and Gowers [8], and the 1-statement of an asymmetric version (conjectured by Kohayakawa and Kreuter [20] in 1997) was recently proved by Mousset, Nenadov, and Samotij [31] (see [17,20,25,27] for progress in the 0-statement).…”

Section: Introductionmentioning

confidence: 87%

Ramsey goodness of trees in random graphs

Araújo

Moreira

Pavez-Signé

2022

Random Struct Algorithms

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For graphs G,H$$ G,H $$ and a family of graphs ℱ$$ \mathcal{F} $$, we write G→false(H,ℱfalse)$$ G\to \left(H,\mathcal{F}\right) $$ to denote that every blue‐red coloring of the edges of G$$ G $$ contains either a blue copy of H$$ H $$, or a red copy of each F∈ℱ$$ F\in \mathcal{F} $$. For integers n$$ n $$ and D$$ D $$, let 𝒯(n,D) denote the family of all trees with n$$ n $$ edges and maximum degree at most D$$ D $$. We prove that for each r,D⩾2$$ r,D\geqslant 2 $$, there exist constants C,C′>0$$ C,{C}^{\prime }>0 $$ such that if p⩾Cnprefix−2false/false(r+2false)$$ p\geqslant C{n}^{-2/\left(r+2\right)} $$ and N⩾rn+C′false/p$$ N\geqslant rn+{C}^{\prime }/p $$, then G(N,p)→(Kr+1,𝒯(n,D))with high probability. This is a random version of a well‐known result of Chvátal from 1977. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erdős–Sós conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.

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Asymmetric Ramsey properties of random graphs involving cliques and cycles

Cited by 2 publications

References 18 publications

A resolution of the Kohayakawa--Kreuter conjecture for the majority of cases

A resolution of the Kohayakawa--Kreuter conjecture for the majority of cases

Ramsey goodness of trees in random graphs

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