Long paths and cycles in random subgraphs of H-free graphs

Let H be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let G be an arbitrary finite H-free graph with minimum degree at least k. For p ∈ [0, 1], we form a p-random subgraph G_p of G by independently keeping each edge of G with probability p. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive ε, there exists a positive δ (depending only on ε) such that the following holds: If p ≥ 1+ε/k, then with probability tending to 1 as k → ∞, the random graph G_p contains a cycle of length at least n_H(δk), where n_H(k) > k is the minimum number of vertices in an H-free graph of average degree at least k. Thus in particular G_p as above typically contains a cycle of length at least linear in k.