{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:I3L2IIPS5JHHW55C4ZA6OJNIMK","short_pith_number":"pith:I3L2IIPS","schema_version":"1.0","canonical_sha256":"46d7a421f2ea4e7b77a2e641e725a862977c0d4bb5dbe681d9b49716ceb40917","source":{"kind":"arxiv","id":"2607.09049","version":1},"attestation_state":"computed","paper":{"title":"A Single-Exponential Erd\\H{o}s--Hajnal Bound for Graphs of Bounded VC-Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiasheng Zeng, Shuang Sun, Yan Wang","submitted_at":"2026-07-10T02:38:07Z","abstract_excerpt":"A homogeneous set in a graph is a clique or a stable set. The Erd\\H{o}s--Hajnal conjecture states that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a homogeneous set of size at least $n^c$. Nguyen, Scott and Seymour proved that for every $d>0$, graphs of VC-dimension at most $d$ have the Erd\\H{o}s--Hajnal property, confirming a conjecture of Fox, Pach and Suk. In particular, they showed that every such $n$-vertex graph contains a homogeneous set of size at least $n^{\\eta_d}$ for some $\\eta_d\\ge 2^{-2^{O(d)}}$. In this paper, we give a sharper quan"},"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":"2607.09049","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-10T02:38:07Z","cross_cats_sorted":[],"title_canon_sha256":"4abaef3b00e8d6a50cdef67339784bfb64612bb18b0c86ef37f6f5db534effe3","abstract_canon_sha256":"295121e78bc590adb33e3f5aaa29d1e67aec10861b7c47f3ab9ab50da43ee01d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-13T00:17:37.692438Z","signature_b64":"P9l0vsmJs81zq//ep/MfaSxEtz2+9tf6zXuQoR0XsAIh0SgVGY31SO7uxAD+o8NJlU4KBKc+wdRSmQL3akARDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46d7a421f2ea4e7b77a2e641e725a862977c0d4bb5dbe681d9b49716ceb40917","last_reissued_at":"2026-07-13T00:17:37.691105Z","signature_status":"signed_v1","first_computed_at":"2026-07-13T00:17:37.691105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Single-Exponential Erd\\H{o}s--Hajnal Bound for Graphs of Bounded VC-Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiasheng Zeng, Shuang Sun, Yan Wang","submitted_at":"2026-07-10T02:38:07Z","abstract_excerpt":"A homogeneous set in a graph is a clique or a stable set. The Erd\\H{o}s--Hajnal conjecture states that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a homogeneous set of size at least $n^c$. Nguyen, Scott and Seymour proved that for every $d>0$, graphs of VC-dimension at most $d$ have the Erd\\H{o}s--Hajnal property, confirming a conjecture of Fox, Pach and Suk. In particular, they showed that every such $n$-vertex graph contains a homogeneous set of size at least $n^{\\eta_d}$ for some $\\eta_d\\ge 2^{-2^{O(d)}}$. In this paper, we give a sharper quan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09049","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.09049/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.09049","created_at":"2026-07-13T00:17:37.691780+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.09049v1","created_at":"2026-07-13T00:17:37.691780+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.09049","created_at":"2026-07-13T00:17:37.691780+00:00"},{"alias_kind":"pith_short_12","alias_value":"I3L2IIPS5JHH","created_at":"2026-07-13T00:17:37.691780+00:00"},{"alias_kind":"pith_short_16","alias_value":"I3L2IIPS5JHHW55C","created_at":"2026-07-13T00:17:37.691780+00:00"},{"alias_kind":"pith_short_8","alias_value":"I3L2IIPS","created_at":"2026-07-13T00:17:37.691780+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/I3L2IIPS5JHHW55C4ZA6OJNIMK","json":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK.json","graph_json":"https://pith.science/api/pith-number/I3L2IIPS5JHHW55C4ZA6OJNIMK/graph.json","events_json":"https://pith.science/api/pith-number/I3L2IIPS5JHHW55C4ZA6OJNIMK/events.json","paper":"https://pith.science/paper/I3L2IIPS"},"agent_actions":{"view_html":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK","download_json":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK.json","view_paper":"https://pith.science/paper/I3L2IIPS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.09049&json=true","fetch_graph":"https://pith.science/api/pith-number/I3L2IIPS5JHHW55C4ZA6OJNIMK/graph.json","fetch_events":"https://pith.science/api/pith-number/I3L2IIPS5JHHW55C4ZA6OJNIMK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK/action/storage_attestation","attest_author":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK/action/author_attestation","sign_citation":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK/action/citation_signature","submit_replication":"https://pith.science/pith/I3L2IIPS5JHHW55C4ZA6OJNIMK/action/replication_record"}},"created_at":"2026-07-13T00:17:37.691780+00:00","updated_at":"2026-07-13T00:17:37.691780+00:00"}