Cycle Ramsey numbers for random graphs

TL;DR

This paper investigates cycle Ramsey numbers in random graphs, establishing asymptotic conditions under which certain cycle colorings are guaranteed, thus strengthening known results with probabilistic methods.

Contribution

It extends cycle Ramsey number results to random graphs, providing new asymptotic thresholds and probabilistic guarantees for cycle colorings.

Findings

Asymptotic Ramsey properties for cycles in random graphs.

Thresholds for cycle colorings in  graphs.

Results for both three-color and two-color cases.

Abstract

Let $C_{n}$ be a cycle of length $n$. As an application of Szemer\'{e}di's regularity lemma, {\L}uczak ($R(C_n,C_n,C_n)\leq (4+o(1))n$, J. Combin. Theory Ser. B, 75 (1999), 174--187) in fact established that $K_{(8+o(1))n}\to(C_{2n+1},C_{2n+1},C_{2n+1})$. In this paper, we strengthen several results involving cycles. Let $\mathcal{G}(n,p)$ be the random graph. We prove that for fixed $0<p\le1$, and integers $n_1$, $n_2$ and $n_3$ with $n_1\ge n_2\ge n_3$, it holds that for any sufficiently small $\delta>0$, there exists an integer $n_0$ such that for all integer $n_3>n_0$, we have a.a.s. that \begin{align*} \mathcal{G}((8+\delta)n_1,p) \to (C_{2n_1+1},C_{2n_2+1},C_{2n_3+1}). \end{align*} Moreover, we prove that for fixed $0<p\le1$ and integers $n_1\ge n_2\ge n_3>0$ with same order, i.e. $n_2=\Theta(n_1)$ and $n_3=\Theta(n_1)$, we have a.a.s. that \begin{align*} \mathcal{G}(2n_1+n_2+n_3+o(1)n_1,p) \to (C_{2n_1},C_{2n_2},C_{2n_3}). \end{align*} Similar results for the two color case are also obtained.