For every fixed graph H and every fixed 0 < α < 1, we show 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 random graph G(n,p); that is, it is p-quasi-random in the sense of Chung, Graham, and Wilson . This solves a conjecture raised by Shapira  and solves in a strong sense an open problem of Simonovits and Sós .
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics