A degree version of the Burr-Erdős conjecture on trees

Abstract

An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr--Erdős conjecture holds. More precisely, Schelp asked if is it true that for any $\epsilon >0$ and $\Delta \ge 2$, if $G$ is a graph on $N\ge (2+\epsilon )n$ vertices and minimum degree $\delta (G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $\Delta$. We prove this conjecture in a strong form, showing that it is true even if one removes the extra $\epsilon n$ term in the size of the host graph.