Clique factors in pseudorandom graphs

An n-vertex graph is said to to be (p, )-bijumbled if for any vertex sets A, B C V(G), we have p e(A, B subset of = V(G) , e(A, B) = p|A|B| +/- beta root|A| |B|. We prove that for any r is an element of N >= 3 and c > 0 there exists an e > 0 such that any n-vertex (p, beta)- bijumbled graph with n is an element of rN, p > 0, delta(G) > cpn and <= epsilon p(r-1)n contains a K-r-factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, lambda)-graphs. For the case of triangle factors, that is, when r = 3, this result resolves a conjecture of Krivelevich, Sudakov and Szab & oacute; from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of beta = o(p(2)n) actually guarantees that a (p, beta)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2.