The generalized Turan number ex(G,H) of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph K"m on m vertices, the value of ex(K"m,H) is (1-1/(@g(H)-1)+o(1))(m2), where o(1)->0 as m->~, by the Erdos-Stone-Simonovits theorem. In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs '85, Poznan, 1985, North-Holland, Amsterdam, 1987, pp. 307-331. MR 89d:05158]. A graph G is (q,@b)-bi-jumbled if|e"G(X,Y)-q|X||Y||= 0 there exists @c>0 so that the following holds: any large enough m-vertex, (q,@cq^@D^+^1^/^2m)-bi-jumbled graph G satisfiesex(G,H)=<(1-1@g(H)-1+@d)|E(G)|.
['Yoshiharu Kohayakawa', 'Vojtěch Rödl', 'Mathias Schacht', 'Papa Sissokho', 'Jozef Skokan']
Turán's theorem for pseudo-random graphs