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The generalized vertex Folkman number $F_v(H_1, H_2; H)$ is defined as the smallest integer $n$ for which there exists an $H$-free graph $G$ of order $n$ such that $G \\rightarrow (H_1, H_2)^v$. The generalized edge Folkman numbers $F_e(H_1, H_2; H)$ are defined similarly, when colorings of the edges are considered.\n  We show that $F_e(K_{k+1},K_{k+1};K_{k+2}-e)$ and $F_v(K_k,K_k;K_{k+1}-e)$ are well defined for $k \\geq 3$"},"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":"1705.06268","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T17:27:05Z","cross_cats_sorted":[],"title_canon_sha256":"f9caab8e2c1f0e00066d87b055c5dbc63b9fccec38ff75d98a77b90728983047","abstract_canon_sha256":"56039282ca1892ab23af1b46df1b4d6b43c982ef12adef55cd0a8d83d0875206"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:48.851201Z","signature_b64":"yDq0ggnGZ0yPr8jSX6mbfUQ7qn0alU0bsH9s6FmljTxEK6HNtp4r8YgEPrB35DrT0d7MGsoUpdCF7Owuq/HUDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec42a3ca0b7f65a3912fb7970f7b97f4198b483ec7ff947d10023df6e359f77a","last_reissued_at":"2026-05-18T00:12:48.850632Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:48.850632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Nonexistence of Some Generalized Folkman Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Meilian Liang, Stanis{\\l}aw Radziszowski, Xiaodong Xu","submitted_at":"2017-05-17T17:27:05Z","abstract_excerpt":"For an undirected simple graph $G$, we write $G \\rightarrow (H_1, H_2)^v$ if and only if for every red-blue coloring of its vertices there exists a red $H_1$ or a blue $H_2$. The generalized vertex Folkman number $F_v(H_1, H_2; H)$ is defined as the smallest integer $n$ for which there exists an $H$-free graph $G$ of order $n$ such that $G \\rightarrow (H_1, H_2)^v$. The generalized edge Folkman numbers $F_e(H_1, H_2; H)$ are defined similarly, when colorings of the edges are considered.\n  We show that $F_e(K_{k+1},K_{k+1};K_{k+2}-e)$ and $F_v(K_k,K_k;K_{k+1}-e)$ are well defined for $k \\geq 3$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.06268","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1705.06268","created_at":"2026-05-18T00:12:48.850713+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.06268v2","created_at":"2026-05-18T00:12:48.850713+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.06268","created_at":"2026-05-18T00:12:48.850713+00:00"},{"alias_kind":"pith_short_12","alias_value":"5RBKHSQLP5S2","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_16","alias_value":"5RBKHSQLP5S2HEJP","created_at":"2026-05-18T12:31:00.734936+00:00"},{"alias_kind":"pith_short_8","alias_value":"5RBKHSQL","created_at":"2026-05-18T12:31:00.734936+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q","json":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q.json","graph_json":"https://pith.science/api/pith-number/5RBKHSQLP5S2HEJPW6LQ664X6Q/graph.json","events_json":"https://pith.science/api/pith-number/5RBKHSQLP5S2HEJPW6LQ664X6Q/events.json","paper":"https://pith.science/paper/5RBKHSQL"},"agent_actions":{"view_html":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q","download_json":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q.json","view_paper":"https://pith.science/paper/5RBKHSQL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.06268&json=true","fetch_graph":"https://pith.science/api/pith-number/5RBKHSQLP5S2HEJPW6LQ664X6Q/graph.json","fetch_events":"https://pith.science/api/pith-number/5RBKHSQLP5S2HEJPW6LQ664X6Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q/action/storage_attestation","attest_author":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q/action/author_attestation","sign_citation":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q/action/citation_signature","submit_replication":"https://pith.science/pith/5RBKHSQLP5S2HEJPW6LQ664X6Q/action/replication_record"}},"created_at":"2026-05-18T00:12:48.850713+00:00","updated_at":"2026-05-18T00:12:48.850713+00:00"}