 
Summary: The number of edge colorings with no monochromatic cliques
Noga Alon
J´ozsef Balogh
Peter Keevash
Benny Sudakov §
Abstract
Let F(n, r, k) denote the maximum possible number of distinct edgecolorings of a simple graph
on n vertices with r colors, which contain no monochromatic copy of Kk. It is shown that for
every fixed k and all n > n0(k), F(n, 2, k) = 2tk1(n)
and F(n, 3, k) = 3tk1(n)
, where tk1(n)
is the maximum possible number of edges of a graph on n vertices with no Kk, (determined by
Tur´an's Theorem). The case r = 2 settles a conjecture of Yuster. On the other hand, for every
fixed r > 3 and k > 2, the function F(n, r, k) is exponentially bigger than rtk1(n)
. The proofs are
based on Szemer´edi's regularity lemma together with some additional tools in Extremal Graph
Theory, and provide an example of a precise result proved by applying this lemma.
1 Introduction
Given a graph G, denote by F(G, r, k) the number of distinct edge colorings of G with r colors which
contain no monochromatic copy of Kk, i.e., a complete graph on k vertices. Let
