{"paper":{"title":"Density Hajnal--Szemer\\'{e}di theorem for cliques of size four","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Caiyun Hu, Jianfeng Hou, Xizhi Liu, Yixiao Zhang","submitted_at":"2025-01-01T10:53:18Z","abstract_excerpt":"The celebrated Corr\\'{a}di--Hajnal Theorem~\\cite{CH63} and the Hajnal--Szemer\\'{e}di Theorem~\\cite{HS70} determined the exact minimum degree thresholds for a graph on $n$ vertices to contain $k$ vertex-disjoint copies of $K_r$, for $r=3$ and general $r \\ge 4$, respectively. The edge density version of the Corr\\'{a}di--Hajnal Theorem was established by Allen--B\\\"ottcher--Hladk\\'y--Piguet~\\cite{ABHP15} for large $n$. Remarkably, they determined the four classes of extremal constructions corresponding to different intervals of $k$. They further proposed the natural problem of establishing a densi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.00801","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/2501.00801/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"}