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arXiv · 2607.17831

A note on tree-cycle Ramsey numbers

Ting Huang·Yanbo Zhang·Yaojun Chen
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Abstract

Let $R(T_n,C_m)$ denote the Ramsey number of a tree $T_n$ on $n$ vertices versus a cycle $C_m$ of length $m$. Burr, Erdős, Faudree, Rousseau, and Schelp (1982) asked for the least function $f(m)$ such that $R(T_n,C_m)=2n-1$ for every odd $m\ge 3$ whenever $n\ge f(m)$. They proved that $f(m)\le 756m^{10}$. This bound was later improved to $25m$ by Brennan (2016) and to $4m-8$ by Fan and Lin (2025). In this note, we show that $f(m)\le 2m-4$ by using a different method and conjecture that $f(m)=\lceil (2m-1)/3\rceil$.

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Ting Huang, Yanbo Zhang, Yaojun Chen. 2026-07-20. A note on tree-cycle Ramsey numbers. https://arxiv.org/abs/2607.17831

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