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Path Ramsey Number for Random Graphs

Published online by Cambridge University Press:  07 December 2015

SHOHAM LETZTER*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: s.letzter@dpmms.cam.ac.uk)

Abstract

Answering a question raised by Dudek and Prałat, we show that if pn → ∞, 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 $(1-\varepsilon)\binom{n}{2}$ edges, whenever G is 2-edge-coloured, there is a monochromatic path of length at least $(2/3 - 110\sqrt{\varepsilon})n$. This is an extension of the classical result by Gerencsér and Gyárfás which says that whenever Kn is 2-coloured there is a monochromatic path of length at least 2n/3.

MSC classification

Type
Paper
Copyright
Copyright © Cambridge University Press 2015 

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