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

For 0 < γ ≤ 1 and graphs G and H, we write G[formula]H if any γ-proportion of the edges of G span at least one copy of H in G. As customary, we write Ck for a cycle of length k. We show that, for every fixed integer l ≥ 2 and real γ > 0, there exists constant C = C(l, γ) > 0 such that almost every random graph Gn, p with p = p(n) ≥ Cn−1 + 1/(2l − 1) satisfies Gn,p[formula]C2l. In particular, for any fixed l ≥ 2 and γ > 0, this result implies the existence of very sparse graphs G with G[formula]C2l.