Threshold functions for asymmetric Ramsey properties involving cycles

Abstract

ABSTRACT:We consider the binomial random graph G and determine a sharp threshold p function for the edge-Ramsey propertyp for all l , . . . , l , where C l denotes the cycle of length l. As deterministic consequences of 1 r our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above.Ž .

Asymmetric Ramsey properties of random graphs involving cliques

“…In the context of asymmetric Ramsey properties, the following generalization of d 2 with two arguments was introduced in [9]. Let H and F be any graphs, and define…”

“…In [9] Kohayakawa and Kreuter also formulated the following conjecture that generalizes their result to general graphs.…”

“…The way of proving it that was pursued in [17] and [9] is by contradiction. This approach shows the existence of a coloring, but provides no efficient way of obtaining the coloring from the proof.…”

“…We provide a (polynomial-time) algorithm that computes a valid coloring for graphs that satisfy certain properties. We employ techniques similar to those in [17] and [9] in order to prove that these properties a.a.s. hold in G n,p with p sufficiently small.…”

Independent sets in hypergraphs

“…There had been little progress on Conjecture 1.10 until quite recently, when the 0-statement was proved by Marciniszyn, Skokan, Spöhel, and Steger [46] in the case where all of the H i are cliques, and the 1-statement in the case r = 2 was established by Kohayakawa, Schacht, and Spöhel [44] under very mild extra assumptions on H 1 and H 2 . Using Theorem 1.8, the approach of Kohayakawa and Kreuter [39], which employs the sparse regularity lemma, can be adapted (see [46,Theorem 31]) to yield a proof of the 1-statement in Conjecture 1.10 for the following class of graphs. Theorem 1.11.…”