Hereditary quasirandomness without regularity

A result of Simonovits and Sos states that for any fixed graph H and any epsilon > 0 there exists delta > 0 such that if G is an n-vertex graph with the property that every S subset of V(G) contains p(e( H)) vertical bar S vertical bar(v(H)) +/- delta n(v(H)) labelled copies of H, then G is quasirandom in the sense that every S subset of V(G) contains 1/2 p vertical bar S vertical bar(2) +/- epsilon n(2) edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on delta(-1) which is a tower of twos of height polynomial in epsilon(-1). We give an alternative proof of this theorem which avoids the regularity lemma and shows that d may be taken to be linear in epsilon when H is a clique and polynomial in epsilon for general H. This answers a problem raised by Simonovits and Sos.