Abstract

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Gamma the generalized Turan density pi(F) (Gamma) denotes the relative density of a maximum subgraph of Gamma, which contains no copy of F. Extending classical Turan type results for odd cycles, we show that pi(F) (Gamma)=1/2 provided F is an odd cycle and Gamma is a sufficiently pseudorandom graph. In particular, for (n,d,lambda)-graphs Gamma, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval [-lambda,lambda], our result holds for odd cycles of length l, provided lambda(l-2) dl-1/nlog(n)(-(l-2)(l-3)). Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szab, and Vu, who addressed the case when F is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free (n,d;lambda)-graphs) shows that our assumption on Gamma is best possible up to the polylog-factor for every odd l >= 5.