A canonical Ramsey theorem for even cycles in random graphs

TL;DR

This paper establishes the threshold probability for the random graph G(n,p) to almost surely contain canonical copies of even cycles under any edge colouring, extending classical Ramsey results to sparse random graphs.

Contribution

It determines the asymptotic threshold for the canonical Ramsey property for even cycles in G(n,p), up to a logarithmic factor, which was previously unknown.

Findings

Threshold for G(n,p) to have canonical even cycles identified

Almost sure presence of canonical cycles in sparse random graphs proven

Extends classical Ramsey theory to probabilistic setting

Abstract

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for $2\leq k\in \mathbb{N}$, any colouring of the edges of $K_n$ with $n$ sufficiently large gives a copy of $C_{2k}$ which has one of three canonical colour patterns: monochromatic, rainbow or lexicographic. In this paper we show that if $p=\omega(n^{-1+1/(2k-1)}\log n)$, then ${\mathbf{G}}(n,p)$ will asymptotically almost surely also have the property that any colouring of its edges induces canonical copies of $C_{2k}$. This determines the threshold for the canonical Ramsey property with respect to even cycles, up to a $\log$ factor.