Extremal results for odd cycles in sparse pseudorandom graphs

We consider extremal problems for subgraphs of pseudorandom graphs. Our results implies that for ( n , d , λ )-graphs Γ satisfying λ 2 k − 1 ≪ d 2 k n ( log n ) − 2 ( k − 1 ) ( 2 k − 1 ) any subgraph G ⊂ Γ not containing a cycle of length 2 k + 1 has relative density at most 1 2 + o ( 1 ) . Up to the polylog-factor the condition on λ is best possible and was conjectured by Krivelevich, Lee and Sudakov.