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We prove this in every fixed polynomial edge-density regime: for all $r\\ge4$, $k\\ge2$, $P,C>0$, there is $M=M_{r,k}(P,C)$ such that $\\chi(G)\\ge M,\\ e(G)\\le C\\chi(G)^P\\Longrightarrow h_r(G)\\ge k.$ Quantitatively, after replacing $P$ by $P\\vee2$ and $C$ by $C\\vee2$, $M_{r,k}(P,C)\\le \\exp!\\left(O_{r,k}\\bigl((P+2+\\log(C\\vee2))^2\\bigr)\\right),$ and consequently the same conclusion holds throughout the quasi-polynom"},"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":"2606.17901","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-16T13:28:17Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"8b4ccd9fb549a9b591b89188c64e997e2d99da25d2f6356ef75eb05714f8cfbf","abstract_canon_sha256":"e8b0c35659a2a0507b5aee3afc446d61299e9ede38b42a0c97bfe3c108cc3f8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:43.136692Z","signature_b64":"u7iGvyI40sOREMoXfztvJAtg3CCL1D2gNMP+9B2U8EzNNdx8yL/L33YTj4jTW/Dal+DSeyP91fo0v5deXQPZAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df87a82fb23f3ed04d847cff7893413b3478968faf5c398dd63b2ea252462976","last_reissued_at":"2026-06-19T16:10:43.136276Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:43.136276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Erd\\H{o}s-Hajnal High-Girth Subgraph Conjecture Holds in the Polynomial Chromatic-Sparsity Regime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-16T13:28:17Z","abstract_excerpt":"For a graph $G$ put $h_r(G)=\\max{\\chi(H):H\\subseteq G,\\operatorname{girth}(H)\\ge r}.$ Erd\\H{o}s and Hajnal asked whether $h_r(G)\\to\\infty$ as $\\chi(G)\\to\\infty$, for every fixed $r\\ge4$. We prove this in every fixed polynomial edge-density regime: for all $r\\ge4$, $k\\ge2$, $P,C>0$, there is $M=M_{r,k}(P,C)$ such that $\\chi(G)\\ge M,\\ e(G)\\le C\\chi(G)^P\\Longrightarrow h_r(G)\\ge k.$ Quantitatively, after replacing $P$ by $P\\vee2$ and $C$ by $C\\vee2$, $M_{r,k}(P,C)\\le \\exp!\\left(O_{r,k}\\bigl((P+2+\\log(C\\vee2))^2\\bigr)\\right),$ and consequently the same conclusion holds throughout the quasi-polynom"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17901","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/2606.17901/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":"2606.17901","created_at":"2026-06-19T16:10:43.136346+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.17901v1","created_at":"2026-06-19T16:10:43.136346+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.17901","created_at":"2026-06-19T16:10:43.136346+00:00"},{"alias_kind":"pith_short_12","alias_value":"36D2QL5SH47N","created_at":"2026-06-19T16:10:43.136346+00:00"},{"alias_kind":"pith_short_16","alias_value":"36D2QL5SH47NATME","created_at":"2026-06-19T16:10:43.136346+00:00"},{"alias_kind":"pith_short_8","alias_value":"36D2QL5S","created_at":"2026-06-19T16:10:43.136346+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/36D2QL5SH47NATMEPT7XRE2BHM","json":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM.json","graph_json":"https://pith.science/api/pith-number/36D2QL5SH47NATMEPT7XRE2BHM/graph.json","events_json":"https://pith.science/api/pith-number/36D2QL5SH47NATMEPT7XRE2BHM/events.json","paper":"https://pith.science/paper/36D2QL5S"},"agent_actions":{"view_html":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM","download_json":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM.json","view_paper":"https://pith.science/paper/36D2QL5S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.17901&json=true","fetch_graph":"https://pith.science/api/pith-number/36D2QL5SH47NATMEPT7XRE2BHM/graph.json","fetch_events":"https://pith.science/api/pith-number/36D2QL5SH47NATMEPT7XRE2BHM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM/action/storage_attestation","attest_author":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM/action/author_attestation","sign_citation":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM/action/citation_signature","submit_replication":"https://pith.science/pith/36D2QL5SH47NATMEPT7XRE2BHM/action/replication_record"}},"created_at":"2026-06-19T16:10:43.136346+00:00","updated_at":"2026-06-19T16:10:43.136346+00:00"}