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arXiv · 2411.14566

A canonical Ramsey theorem for even cycles in random graphs

José D. Alvarado·Y. Kohayakawa·Patrick Morris·Guilherme O. Mota
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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.

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José D. Alvarado, Y. Kohayakawa, Patrick Morris, Guilherme O. Mota. 2024-11-21. A canonical Ramsey theorem for even cycles in random graphs. https://arxiv.org/abs/2411.14566

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