On the Multi-coloured Ramsey Numbers of Cycles

TL;DR

This paper investigates the multi-coloured Ramsey numbers of cycles, proving asymptotic upper bounds for odd and even cycles for any number of colours, advancing understanding of colourings in complete graphs.

Contribution

It establishes new asymptotic upper bounds for the multi-coloured Ramsey numbers of cycles for all numbers of colours, including cases with large odd and even cycles.

Findings

For odd cycles, R_k(C_n) ≤ k2^k n + o(n) as n→∞.

For even cycles, R_k(C_n) ≤ kn + o(n) as n→∞.

Progress on conjectures for multi-coloured Ramsey numbers of cycles.

Abstract

For a graph $L$ and an integer $k\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$ as a subgraph. Bondy and Erd\H{o}s conjectured that for an odd cycle $C_n$ on $n$ vertices, $$R_k(C_n) = 2^{k-1}(n-1)+1 \text{for $n>3$.}$$ They proved the case when $k=2$ and also provided an upper bound $R_k(C_n)\leq (k+2)!n$. Recently, this conjecture has been verified for $k=3$ if $n$ is large. In this note, we prove that for every integer $k\geq 4$, $$R_k(C_n)\leq k2^kn+o(n), \text{as $n\to\infty$.}$$ When $n$ is even, Yongqi, Yuansheng, Feng, and Bingxi gave a construction, showing that $R_k(C_n)\geq (k-1)n-2k+4.$ Here we prove that if $n$ is even, then $$R_k(C_n)\leq kn+o(n), \text{as $n\to\infty$.}$$