Hereditary quasirandomness without regularity

Copyright © Cambridge Philosophical Society 2017 A result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains p e(H) |S| v(H) ± δ n v(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains (Formula presented.) p|S| 2 ± ε n 2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ −1 which is a tower of twos of height polynomial in ε −1 . We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.