{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:36D2QL5SH47NATMEPT7XRE2BHM","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":"e8b0c35659a2a0507b5aee3afc446d61299e9ede38b42a0c97bfe3c108cc3f8d","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-16T13:28:17Z","title_canon_sha256":"8b4ccd9fb549a9b591b89188c64e997e2d99da25d2f6356ef75eb05714f8cfbf"},"schema_version":"1.0","source":{"id":"2606.17901","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.17901","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.17901v1","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.17901","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"pith_short_12","alias_value":"36D2QL5SH47N","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"pith_short_16","alias_value":"36D2QL5SH47NATME","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"pith_short_8","alias_value":"36D2QL5S","created_at":"2026-06-19T16:10:43Z"}],"graph_snapshots":[{"event_id":"sha256:ea30343b972549a7a08849792ab40d0934c974628fe7cf4fac4dee335f48faa5","target":"graph","created_at":"2026-06-19T16:10:43Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.17901/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Eric Li (Trinity College, University of Cambridge)","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-16T13:28:17Z","title":"The Erd\\H{o}s-Hajnal High-Girth Subgraph Conjecture Holds in the Polynomial Chromatic-Sparsity Regime"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17901","kind":"arxiv","version":1},"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":"sha256:9fcf2d67899de24baccc76b4233837c28714338ae267849f87a11abc83c1489e","target":"record","created_at":"2026-06-19T16:10:43Z","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":"e8b0c35659a2a0507b5aee3afc446d61299e9ede38b42a0c97bfe3c108cc3f8d","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-16T13:28:17Z","title_canon_sha256":"8b4ccd9fb549a9b591b89188c64e997e2d99da25d2f6356ef75eb05714f8cfbf"},"schema_version":"1.0","source":{"id":"2606.17901","kind":"arxiv","version":1}},"canonical_sha256":"df87a82fb23f3ed04d847cff7893413b3478968faf5c398dd63b2ea252462976","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"df87a82fb23f3ed04d847cff7893413b3478968faf5c398dd63b2ea252462976","first_computed_at":"2026-06-19T16:10:43.136276Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:10:43.136276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u7iGvyI40sOREMoXfztvJAtg3CCL1D2gNMP+9B2U8EzNNdx8yL/L33YTj4jTW/Dal+DSeyP91fo0v5deXQPZAg==","signature_status":"signed_v1","signed_at":"2026-06-19T16:10:43.136692Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.17901","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9fcf2d67899de24baccc76b4233837c28714338ae267849f87a11abc83c1489e","sha256:ea30343b972549a7a08849792ab40d0934c974628fe7cf4fac4dee335f48faa5"],"state_sha256":"b7f60c943734c54377f9e4c664b7bfb25abb6023ffbd93153c37359ef18f5e84"}