A Note on Tree–cycle Ramsey Numbers

Let R ( T n , C m ) R(Tₙ,Cₘ) 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 Tₙ upper T Subscript n on n n n vertices versus a cycle C m Cₘ 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(Tₙ,Cₘ)=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