Path Ramsey number for random graphs

TL;DR

This paper proves that in random graphs with certain edge probabilities, any 2-coloring guarantees a long monochromatic path of length approximately two-thirds of the vertices, extending classical results to sparse graphs.

Contribution

It establishes the existence of near-two-thirds length monochromatic paths in random graphs and extends classical results to graphs with high edge density.

Findings

Monochromatic path length is approximately 2n/3 in G(n,p) with p*n→∞.

In dense graphs, monochromatic paths of length at least (2/3 - 100√ε)n exist.

The results are optimal; 2/3 cannot be improved as a constant.

Abstract

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. 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 which may be of independent interest. We show that given a graph $G$ on $n$ vertices with at least $(1-\epsilon)\binom{n}{2}$ edges, whenever $G$ is $2$-edge-coloured, there is a monochromatic path of length at least $(2/3-100\sqrt{\epsilon})n$. This is an extension of the classical result by Gerencs\'er and Gy\'arf\'as which says that whenever $K_n$ is $2$-coloured there is a monochromatic path of length at least $2n/3$.