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Here we verify this for $k \\leq C\\log n$ with any fixed $C>0$. In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for $k\\geq \\kappa_0 \\log n$ (with $\\kappa_0\\approx 4.82$)."},"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":"1401.2710","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-13T04:43:00Z","cross_cats_sorted":[],"title_canon_sha256":"9464e785c5c3f3c905b9b3fe458eb80ac9932c085a4e822c942592a49b10a48d","abstract_canon_sha256":"542b67f3828486d8bf0ab663f11cf26c0153e33f46e0d997ffc0364fa6c4fb02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:32.473240Z","signature_b64":"wU2smhwehSJIQqAZh1N6JEPoa0fd87nrfqNUjCAJ79avx998IcHEHgvtNSiB/D3JNWyzKO6pTyORLheLQ5WpDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20a444fe26d586c33897307acc3c434e4d361cd723e1d04fe85d24e6d34382a1","last_reissued_at":"2026-05-18T03:02:32.472492Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:32.472492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The threshold for combs in random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eyal Lubetzky, Jeff Kahn, Nicholas Wormald","submitted_at":"2014-01-13T04:43:00Z","abstract_excerpt":"For $k\\mid n$ let $Comb_{n,k}$ denote the tree consisting of an $(n/k)$-vertex path with disjoint $k$-vertex paths beginning at each of its vertices. 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