On some Multicolor Ramsey Properties of Random Graphs

The size-Ramsey number R̂(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 colouring of the edges of G with two colours yields a monochromatic copy of F . In this paper, first we focus on the size-Ramsey number of a path Pn on n vertices. In particular, we show that 5n/2 − 15/2 ≤ R̂(Pn) ≤ 74n for n sufficiently large. (The upper bound uses expansion properties of random d-regular graphs.) This improves the previous lower bound, R̂(Pn) ≥ (1 + √ 2)n−O(1), due to Bollobás, and the upper bound, R̂(Pn) ≤ 91n, due to Letzter. Next we study long monochromatic paths in edge-coloured random graph G(n, p) with pn→∞. Let α > 0 be an arbitrarily small constant. Recently, Letzter showed that a.a.s. any 2-edge colouring of G(n, p) yields a monochromatic path of length (2/3−α)n, which is optimal. Extending this result, we show that a.a.s. any 3-edge colouring of G(n, p) yields a monochromatic path of length (1/2− α)n, which is also optimal. We also consider a related problem and show that for any r ≥ 2, a.a.s. any r-edge colouring of G(n, p) yields a monochromatic connected subgraph on (1/(r − 1) − α)n vertices, which is also tight.