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The Erdős–Rothschild problem on edge-colourings with forbidden monochromatic cliques

Published online by Cambridge University Press:  23 January 2017

OLEG PIKHURKO ,
KATHERINE STADEN  and
ZELEALEM B. YILMA
OLEG PIKHURKO
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL. e-mail: o.pikhurko@warwick.ac.uk, k.l.staden@warwick.ac.uk
KATHERINE STADEN
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL. e-mail: o.pikhurko@warwick.ac.uk, k.l.staden@warwick.ac.uk
ZELEALEM B. YILMA
Affiliation:
Carnegie Mellon University Qatar, Doha, Qatar. e-mail: zyilma@qatar.cmu.edu
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Abstract

Let k := (k1,. . .,ks) be a sequence of natural numbers. For a graph G, let F(G;k) denote the number of colourings of the edges of G with colours 1,. . .,s such that, for every c ∈ {1,. . .,s}, the edges of colour c contain no clique of order kc. Write F(n;k) to denote the maximum of F(G;k) over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases.

We prove that, for every k and n, there is a complete multipartite graph G on n vertices with F(G;k) = F(n;k). Also, for every k we construct a finite optimisation problem whose maximum is equal to the limit of log2F(n;k)/${n\choose 2}$ as n tends to infinity. Our final result is a stability theorem for complete multipartite graphs G, describing the asymptotic structure of such G with F(G;k) = F(n;k) · 2o(n2) in terms of solutions to the optimisation problem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 2 , September 2017 , pp. 341 - 356
Copyright
Copyright © Cambridge Philosophical Society 2017 

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