A canonical Ramsey theorem for even cycles in random graphs

2024/11/21 by José D. Alvarado, Alvarado, José D., Yoshiharu Kohayakawa +5 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2411.14566

openalex publication_date 2024/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The celebrated canonical Ramsey theorem of Erdős and Rado implies that for 2≤ k∈ 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=ω(n^-1+1/(2k-1)log n), then 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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