Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least en 2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f ðe; HÞn h copies of H: This is proved by establishing a directed version of Szemere´di's regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of e only for testing the property P H of being

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... As is common with applications of the undirected regularity lemma, here too the function 1=f ðe; HÞ is an extremely fast growing function in e: We therefore further prove a precise characterization of all the digraphs H; for which f ðe; HÞ has a polynomial dependency on e: This implies a characterization of all the digraphs H; for which the property of being H-free has a one-sided error property tester whose query complexity is polynomial in 1=e: We further show that the same characterization also applies to two-sided error property testers as well. A special case of this result settles an open problem raised by the first author in (Alon, Proceedings of the 42nd IEEE FOCS, IEEE, New York, 2001, pp. 434-441). Interestingly, it turns out that if P H has a polynomial query complexity, then there is a two-sided e-tester for P H that samples only Oð1=eÞ vertices, whereas any one-sided tester for P H makes at least ð1=eÞ d=2 queries, where d is the average degree of H: We also show that the complexity of deciding if for a given directed graph H; P H has a polynomial query complexity, is NP-complete, marking an interesting distinction from the case of undirected graphs. (N. Alon), asafico@math.tau.ac.il (A. Shapira).