Upper bounds on probability thresholds for asymmetric Ramsey properties

Abstract

Abstract. Given two graphs G and H, we investigate for which functions p " ppnq the random graph G n,p (the binomial random graph on n vertices with edge probability p) satisfies with probability 1´op1q that every red-blue-coloring of its edges contains a red copy of G or a blue copy of H. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes the exact order of magnitude of the threshold for the case in which G and H are complete graphs of arbitrary size.In our proof we present an alternative to the so-called deletion method, which was introduced by Rödl and Ruciński in their study of symmetric Ramsey properties of random graphs (i.e. the case G " H), and has been used in many proofs of similar results since then. §1. Introduction 1.1. Ramsey properties of random graphs. Ramsey properties of random graphs were studied first by Frankl and Rödl [6], and much effort has been devoted to their further investigation since then. Perhaps most notably, Rödl and Ruciński [20, 21] established a general threshold result that we present in the following.For any two graphs F and H, let F Ñ pHq k denote the property that every edge-coloring of F with k colors contains a monochromatic copy of H. Throughout, we denote the number of edges and vertices of a graph G by e G

Combinatorial theorems in sparse random sets

“…A slightly weaker statement was established by Kohayakawa, Schacht and Spöhel [79] without appealing to the K LR conjecture. Their proof is much closer in spirit to Rödl and Ruciński's proof of Theorem 2.1.…”

A new proof of the KŁR conjecture

“…• Ramsey properties of random graphs: A breakthrough result by Rödl and Ruciński [26] and further extensions to the asymmetric version of the problem [18,22];…”

Extremal Results in Random Graphs

“…In [62] the 1-statement of Conjecture 4.1 for R n (C, F ) for any cycle C and any 2-balanced graph F with m 2 (C) ≥ m 2 (F ) was verified. Moreover, the 0-statement was shown for the case when F 1 and F 2 are cliques [82], and the 1-statement was shown for graphs F 1 and F 2 with m 2 (F 1 , F 2 ) > m 2 (F 1 , F ) for every F F 2 with e(F ) ≥ 1 appeared in [69]. In particular, those results yield the threshold for R(K k , K ).…”