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A NOTE ON TREE–CYCLE RAMSEY NUMBERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 21 September 2026, pp. 1-5
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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 et al. [‘Ramsey numbers for the pair sparse graph–path or cycle’, Trans. Amer. Math. Soc. 269(2) (1982), 501–512] 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 [‘Ramsey numbers of trees versus odd cycles’, Electron. J. Combin. 23(3) (2016), Article no. P3.2] and to $4m-8$
by Fan and Lin [‘Asymptotically optimal Ramsey goodness of sparse graphs versus odd cycles and paths’, Preprint, 2025, arXiv:2507.11835v5]. 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 $
.
