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If $m$ and $p$ are positive integers, then $$ G \\overset{v}{\\rightarrow} {m}\\big\\vert_{p} $$ means that for arbitrary positive integers $a_1, ..., a_s$ ($s$ is not fixed), such that $\\sum_{i = 1}^{s}(a_i - 1) + 1 = m$ an $\\max{\\{a_1, ..., a_s\\}} \\leq p$ we have $G \\overset{v}{\\rightarrow} (a_1, .."},"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":"1511.02125","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-06T15:50:21Z","cross_cats_sorted":[],"title_canon_sha256":"833d7b1f9efd042d93ae00d5efa36da4303d2101764571f1f4cd28f3c00da888","abstract_canon_sha256":"0814c270598804d3741a0fdd7c9ca3aa0eb71b8489fe6940a9f7ad9fcd0dbf9b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:07.711976Z","signature_b64":"ufMkbKQ8klNJgJAH8kJTaAWbx8X3c8HJvLpWUMbdbFmsQb1SOD14ADh2aDUaPWxoGNKnLRA/5cHsVByCKoyjDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73641b4365ad738da5f21d7dc937a0a8f4757323e0ac4cd0a14a94c4332016cc","last_reissued_at":"2026-05-17T23:50:07.711295Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:07.711295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Modified vertex Folkman numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Bikov, Nedyalko Nenov","submitted_at":"2015-11-06T15:50:21Z","abstract_excerpt":"Let $a_1, ..., a_s$ be positive integers. For a graph $G$ the expression $$ G \\overset{v}{\\rightarrow} (a_1, ..., a_s) $$ means that for every coloring of the vertices of $G$ in $s$ colors ($s$-coloring) there exists $i \\in \\{1, ..., s\\}$, such that there is a monochromatic $a_i$-clique of color $i$. If $m$ and $p$ are positive integers, then $$ G \\overset{v}{\\rightarrow} {m}\\big\\vert_{p} $$ means that for arbitrary positive integers $a_1, ..., a_s$ ($s$ is not fixed), such that $\\sum_{i = 1}^{s}(a_i - 1) + 1 = m$ an $\\max{\\{a_1, ..., a_s\\}} \\leq p$ we have $G \\overset{v}{\\rightarrow} (a_1, .."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02125","kind":"arxiv","version":1},"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":"1511.02125","created_at":"2026-05-17T23:50:07.711427+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.02125v1","created_at":"2026-05-17T23:50:07.711427+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.02125","created_at":"2026-05-17T23:50:07.711427+00:00"},{"alias_kind":"pith_short_12","alias_value":"ONSBWQ3FVVZY","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"ONSBWQ3FVVZY3JPS","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"ONSBWQ3F","created_at":"2026-05-18T12:29:34.919912+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/ONSBWQ3FVVZY3JPSDV64SN5AVD","json":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD.json","graph_json":"https://pith.science/api/pith-number/ONSBWQ3FVVZY3JPSDV64SN5AVD/graph.json","events_json":"https://pith.science/api/pith-number/ONSBWQ3FVVZY3JPSDV64SN5AVD/events.json","paper":"https://pith.science/paper/ONSBWQ3F"},"agent_actions":{"view_html":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD","download_json":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD.json","view_paper":"https://pith.science/paper/ONSBWQ3F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.02125&json=true","fetch_graph":"https://pith.science/api/pith-number/ONSBWQ3FVVZY3JPSDV64SN5AVD/graph.json","fetch_events":"https://pith.science/api/pith-number/ONSBWQ3FVVZY3JPSDV64SN5AVD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD/action/storage_attestation","attest_author":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD/action/author_attestation","sign_citation":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD/action/citation_signature","submit_replication":"https://pith.science/pith/ONSBWQ3FVVZY3JPSDV64SN5AVD/action/replication_record"}},"created_at":"2026-05-17T23:50:07.711427+00:00","updated_at":"2026-05-17T23:50:07.711427+00:00"}