# Quasi-randomness and the distribution of copies of a fixed graph

```@article{Shapira2008QuasirandomnessAT,
title={Quasi-randomness and the distribution of copies of a fixed graph},
author={Asaf Shapira},
journal={Combinatorica},
year={2008},
volume={28},
pages={735-745}
}```
• A. Shapira
• Published 1 November 2008
• Mathematics
• Combinatorica
We show that if a graph G has the property that all subsets of vertices of size n/4 contain the “correct” number of triangles one would expect to find in a random graph G(n, 1/2), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sós [10], who showed that in order to deduce that G is quasi-random one needs to assume that all sets of vertices have the correct number of triangles…
Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets
It is shown that if a graph G has the property that all subsets of size αn contain the “correct” number of copies of H one would expect to find in the random graph G(n,p), then G behaves like the randomgraph G( n,p); that is, it is p-quasi-random in the sense of Chung, Graham, and Wilson [4].
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• 2010
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Mathematical Proceedings of the Cambridge Philosophical Society
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• 2013
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