Ramsey goodness of trees in random graphs

Abstract

For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every blue-red colouring of the edges of $G$ contains either a blue copy of $K_{r+1}$, or a red copy of each tree with $n$ edges and maximum degree at most $D$. In 1977, Chv\'atal proved that for any integers $r,n,D \ge 2$, $K_N \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if and only if $N \ge rn+1$. We prove a random analogue of Chv\'atal's theorem for bounded degree trees, that is, we show that for each $r,D\ge 2$ there exist constants $C,C'>0$ such that if $p \ge C{n}^{-2/(r+2)}$ and $N \geq rn + C'/p$, then $$G(N,p) \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$$ with high probability as $n\to \infty$. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erd\H{o}s--S\'os conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.Comment: 30 pages, 3 figure