Turán's Extremal Problem in Random Graphs: Forbidding Even Cycles

For 0 0, there exists a real constantC=C(l, η), such that almost every random graphGn, p withp=p(n)≥Cn−1+1/2l satisfiesGn,p→1/2+ηC2l+1. In particular, for any fixedl≥1 and η>0, this result implies the existence of very sparse graphsG withG→1/2+ηC2l+1.