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Path Ramsey Number for Random Graphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  07 December 2015

SHOHAM LETZTER
SHOHAM LETZTER*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: [email protected])
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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

Primary: 05D10: Ramsey theory
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 4 , July 2016 , pp. 612 - 622
Copyright
Copyright © Cambridge University Press 2015 

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