A NOTE ON TREE–CYCLE RAMSEY NUMBERS

Abstract

Abstract Let R ( T n , C m ) $R(T_n,C_m)$ upper R left parenthesis upper T Subscript n Baseline comma upper C Subscript m Baseline right parenthesis 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