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A chain of length $t$ is a sequence of $t+1$ vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all $n$, there exists $N$ such that every prime graph with at least $N$ vertices contains one of the following graphs or their complements "},"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":"1504.05322","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-21T07:13:31Z","cross_cats_sorted":[],"title_canon_sha256":"b0a72e04b3f6cb9c320765ee7a881639a1941d9e43b2c055f4a4dffcdf57f496","abstract_canon_sha256":"42b124dc1e432fe55c88b4541476417020f74c7d7ec5652390044b0c05cf19d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:38.621484Z","signature_b64":"wGavfhX5bbUqoh5CU0QXfUNMksddz/oIFUCT4QM8w6StaERRqeKp2al4qKdDV3QMSR9Irqg293mswVi1tyvPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88a10200d0e60147388a5ff6ab73278ec25fe9f0d06abb78ba22c3e4f978c975","last_reissued_at":"2026-05-18T01:10:38.620967Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:38.620967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unavoidable induced subgraphs in large graphs with no homogeneous sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Maria Chudnovsky, Paul Seymour, Ringi Kim, Sang-il Oum","submitted_at":"2015-04-21T07:13:31Z","abstract_excerpt":"A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\\le |X|\\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. 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