{"paper":{"title":"On the completion of $\\epsilon$-dense partial Latin squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shikang Yu, Tao Feng","submitted_at":"2026-06-30T05:16:40Z","abstract_excerpt":"A partial Latin square of order $n$ is called $\\epsilon$-dense if each row and each column contains at most $\\epsilon n$ filled cells, and each symbol occurs at most $\\epsilon n$ times. A partial Latin square is said to be completable if its empty cells can be filled to obtain a Latin square. Daykin and H\\\"{a}ggkvist conjectured that every $\\frac{1}{4}$-dense partial Latin square is completable. In this paper, we show that for all sufficiently large integers $n$, every $\\frac{2}{25}$-dense partial Latin square of order $n$ is completable. The proof is obtained by establishing that there exists"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31143","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/2606.31143/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"}