On some Multicolor Ramsey Properties of Random Graphs

Abstract

The size-Ramsey number ${R}{F}$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any coloring of the edges of $G$ with two colors yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le {R}(){P_n} \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This improves the previous lower bound, ${R}{P_n} \ge (1+\sqrt{2})n-O(1)$, due to Bollobas, and the upper bound, ${R}(){P_n} \le 91n$, due to Letzter. Next we study long monochromatic paths in an edge-colored random graph $\mathcal{G}(n,p)$ with $pn \to \infty$. Let $\alpha > 0$ be an arbitrarily small constant. Recently, Letzter showed that asymptotically almost surely (a.a.s.) any $2$-edge coloring of $\mathcal{G}(n,p)$ yields a monochromatic path of length $(2/3-\alpha)n$, which is optimal. Extending this result, we show that ...