Asymmetric Ramsey properties of random graphs involving cliques and cycles

We say thatG→(F,H)if, in every edge coloringc∶E(G)→{1,2}, we can find either a 1-colored copy ofFor a 2-colored copy ofH. The well-known states thatthe threshold for the propertyG(n,p)→(F,H)is equal ton−1∕m2(F,H),wherem2(F,H)is given bym2(F,H)∶=max{e(J)v(J)−2+1∕m2(H)∶J⊆F,e(J)≥1},for any pair of graphsFandHwithm2(F)≥m2(H).In this article, we show the 0-statement of the Kohayakawa–Kreuter conjecture for every pair of cycles and cliques.