Path Ramsey number for random graphs
Shoham Letzter

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 , w.h.p.,~whenever is -coloured, there exists a monochromatic path of length . This result is optimal in the sense that 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 on vertices with at least edges, whenever is -edge-coloured, there is a monochromatic path of length at least . This is an extension of the classical result by Gerencs\'er and Gy\'arf\'as which says that whenever is -coloured there is a monochromatic path of length at least .
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