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

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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 Gₚ 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 Gₚ contains a cycle of length at least nₕ(δk), where nₕ(k)>k is the minimum number of vertices in an H-free graph of average degree at least k. Thus in particular Gₚ as above typically contains a cycle of length at least linear in k.