Degree conditions for Ramsey goodness of paths

Abstract

A classical result of Chv\'atal implies that if $n \geq (r-1)(t-1) +1$, then any colouring of the edges of $K_n$ in red and blue contains either a monochromatic red $K_r$ or a monochromatic blue $P_t$. We study a natural generalization of his result, determining the exact minimum degree condition for a graph $G$ on $n = (r - 1)(t - 1) + 1$ vertices which guarantees that the same Ramsey property holds in $G$. In particular, using a slight generalization of a result of Haxell, we show that $\delta(G) \geq n - \lceil t/2 \rceil$ suffices, and that this bound is best possible. We also use a classical result of Bollob\'as, Erd\H{o}s, and Straus to prove a tight minimum degree condition in the case $r = 3$ for all $n \geq 2t - 1$.