{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:4USCRHB3WZGYJ6GQUFX47VJIRT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d497675bc07eb789182c57247f6be6c023452b96f0c143269fa18ed7f77251b4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-07T17:03:51Z","title_canon_sha256":"2cbbf1f4478a9a81b9d915f858069c26948ece00e9b37bfad045ed1fb943a333"},"schema_version":"1.0","source":{"id":"1703.02473","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.02473","created_at":"2026-05-18T00:41:24Z"},{"alias_kind":"arxiv_version","alias_value":"1703.02473v2","created_at":"2026-05-18T00:41:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.02473","created_at":"2026-05-18T00:41:24Z"},{"alias_kind":"pith_short_12","alias_value":"4USCRHB3WZGY","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"4USCRHB3WZGYJ6GQ","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"4USCRHB3","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:74c755db001b999175a345406cce5a857d4d3d32ae6acbd267d86f65d5f6e2ad","target":"graph","created_at":"2026-05-18T00:41:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Folkman's Theorem asserts that for each $k \\in \\mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \\subset [n]$ of size $k$ with the property that all the sums of the form $\\sum_{x \\in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same colour. In 1989, Erd\\H{o}s and Spencer showed that $F(k) \\ge 2^{ck^2/ \\log k}$, where $c >0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \\ge 2^{2^{k-1}/k}$ for all $k\\in \\mathbb{N}$.","authors_text":"Adam Zsolt Wagner, Andrew Treglown, Bhargav Narayanan, J\\'ozsef Balogh, Sean Eberhard","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-07T17:03:51Z","title":"An improved lower bound for Folkman's theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02473","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:07de0323c60beaed733038f81d9fc2e6a3ba32e68d91df56332a7c64421604b2","target":"record","created_at":"2026-05-18T00:41:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d497675bc07eb789182c57247f6be6c023452b96f0c143269fa18ed7f77251b4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-07T17:03:51Z","title_canon_sha256":"2cbbf1f4478a9a81b9d915f858069c26948ece00e9b37bfad045ed1fb943a333"},"schema_version":"1.0","source":{"id":"1703.02473","kind":"arxiv","version":2}},"canonical_sha256":"e524289c3bb64d84f8d0a16fcfd5288cfe273612143c41d6f6973d46b805dd4a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e524289c3bb64d84f8d0a16fcfd5288cfe273612143c41d6f6973d46b805dd4a","first_computed_at":"2026-05-18T00:41:24.644055Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:24.644055Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tXHG6zNhUeZv2IpBXYACMgjnAtUL3QiV/kzV4iXJi9Ykhf5FCIbWaW3axy294IWdYj4wD+J2NTCd/jmIUqu8CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:24.644725Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.02473","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:07de0323c60beaed733038f81d9fc2e6a3ba32e68d91df56332a7c64421604b2","sha256:74c755db001b999175a345406cce5a857d4d3d32ae6acbd267d86f65d5f6e2ad"],"state_sha256":"ec67f9a3e6a6f9b82888d90d845a8d1d71fe7a1f395ce2c99a09dc5ba568d1af"}