Asymmetric Ramsey Properties of Random Graphs for Cliques and Cycles

TL;DR

This paper proves the zero-probability part of a conjecture regarding the threshold for asymmetric Ramsey properties in random graphs, specifically for pairs of cycles and cliques, advancing understanding in probabilistic combinatorics.

Contribution

It establishes the zero-statement of the Kohayakawa--Kreuter conjecture for all pairs of cycles and cliques, a significant step in asymmetric Ramsey theory.

Findings

Confirmed the zero-probability threshold for all pairs of cycles and cliques.

Validated the conjectured threshold formula for these graph pairs.

Enhanced understanding of asymmetric Ramsey properties in random graphs.

Abstract

We say that $G \to (F,H)$ if, in every edge colouring $c: E(G) \to \{1,2\}$, we can find either a $1$-coloured copy of $F$ or a $2$-coloured copy of $H$. The well-known Kohayakawa--Kreuter conjecture states that the threshold for the property $G(n,p) \to (F,H)$ is equal to $n^{-1/m_{2}(F,H)}$, where $m_{2}(F,H)$ is given by \[ m_{2}(F,H):= \max \left\{\dfrac{e(J)}{v(J)-2+1/m_2(H)} : J \subseteq F, e(J)\ge 1 \right\}. \] In this paper, we show the $0$-statement of the Kohayakawa--Kreuter conjecture for every pair of cycles and cliques.