A NOTE ON TREE–CYCLE RAMSEY NUMBERS

Abstract

Let

denote the Ramsey number of a tree

T n $T_n$ upper T Subscript n

on

n $n$ n

vertices versus a cycle

C m $C_m$ upper C Subscript m

of length

m $m$ 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 ) $f(m)$ f left parenthesis m right parenthesis

such that

R ( T n , C m ) = 2 n − 1 $R(T_n,C_m)=2n-1$ upper R left parenthesis upper T Subscript n Baseline comma upper C Subscript m Baseline right parenthesis equals 2 n minus 1

for every odd

m ≥ 3 $m\ge 3$ m greater than or equals 3

whenever

n ≥ f ( m ) $n\ge f(m)$ n greater than or equals f left parenthesis m right parenthesis

. They proved that

f ( m ) ≤ 756 m 10 $f(m)\le 756m^{10}$ f left parenthesis m right parenthesis less than or equals 756 m Superscript 10

. This bound was later improved to

25 m $25m$ 25 m

by Brennan [‘Ramsey numbers of trees versus odd cycles’, Electron. J. Combin. 23 (3) (2016), Article no. P3.2] and to

4 m − 8 $4m-8$ 4 m minus 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 ) ≤ 2 m − 4 $f(m)\le 2m-4$ f left parenthesis m right parenthesis less than or equals 2 m minus 4

by using a different method and conjecture that

f ( m ) = ⌈ ( 2 m − 1 ) / 3 ⌉ $f(m)=\lceil (2m-1)/3\rceil $ f left parenthesis m right parenthesis equals left ceiling left parenthesis 2 m minus 1 right parenthesis divided by 3 right ceiling

.