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Extending a classical result of Ajtai, Koml\\'os, and Szemer\\'edi, we prove that for every positive $\\varepsilon$, there exists a positive $\\delta$ (depending only on $\\varepsilon$) such that the following holds: If $p \\ge \\frac{1+\\varepsilon}{k}$, then with probability tend"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.1066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-05T15:26:21Z","cross_cats_sorted":[],"title_canon_sha256":"6beb248a1fabd52d607914c3f7cb92a1eacd6de192f64f43aeea8cf20f6ae840","abstract_canon_sha256":"48384556886dd742e4982f207bc5c28135c6167c1c60514a30463e29b4eeed8b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:05.708665Z","signature_b64":"tFNDWpdjo/jkRSsm/20LWI5aXggDtu2dsZbAHiQzYMIatSO/Ipx0WNbiK0TG9JrSKmgKnyNd6I4TyQUu7NQnDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49dddb62aed8bce5605fab283eea9e7b3c387019b2a9c1e6e12184a6481cd081","last_reissued_at":"2026-05-18T03:02:05.708047Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:05.708047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Long paths and cycles in random subgraphs of H-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michael Krivelevich, Wojciech Samotij","submitted_at":"2013-03-05T15:26:21Z","abstract_excerpt":"Let $\\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\\mathcal{H}$-free graph with minimum degree at least $k$. 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