2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 309316
author:  Stefanie Gerke, Martin Marciniszyn and Angelika Steger 

title:  A Probabilistic Counting Lemma for Complete Graphs 
keywords:  
abstract: 
We prove the existence of many complete graphs in almost
all sufficiently dense partitions obtained by an
application of Szemerédi's Regularity Lemma. More
precisely, we consider the number of complete graphs
K
on
ℓ
ℓ
vertices in
ℓ
partite graphs where each partition class consists
of
n
vertices and there is an
ε
regular graph on
m
edges between any two partition classes. We show that
for all
β> 0
, at most a
β
fraction of graphs in this family contain less than
the expected number of copies of
m
K
provided
ℓ
ε
is sufficiently small and
m ≥Cn
for a constant
21/(ℓ1)
C > 0
and
n
sufficiently large. This result is a counting version
of a restricted version of a conjecture by Kohayakawa,
Łuczak and Rödl [MR1479298] and has several
implications for random graphs.

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reference:  Stefanie Gerke and Martin Marciniszyn and Angelika Steger (2005), A Probabilistic Counting Lemma for Complete Graphs, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 309316 
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