{"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"}