Path Ramsey Number for Random Graphs

Answering a question raised by Dudek and Pralat, we show that if pn -> infinity, w.h.p., whenever G = G(n, p) is 2-edge-coloured there is a monochromatic path of length (2/3 + o(1))n. This result is optimal in the sense that 2/3 cannot be replaced by a larger constant. As part of the proof we obtain the following result. Given a graph G on n vertices with at least (1 - epsilon) [GRAPHICS] edges, whenever G is 2-edge-coloured, there is a monochromatic path of length at least (2/3 - 110 root epsilon)n. This is an extension of the classical result by Gerencser and Gyarfas which says that whenever K-n is 2-coloured there is a monochromatic path of length at least 2n/3.