{"paper":{"title":"An Improved Lower Bound for the Erd\\H{o}s-Lov\\'asz Cover Number Problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Varun Sivashankar","submitted_at":"2026-06-23T17:55:10Z","abstract_excerpt":"Let $g(r)$ be the minimum number of edges in an $r$-uniform intersecting hypergraph with cover number $r$. Erd\\H{o}s and Lov\\'asz proved the lower bound $g(r)\\ge 8r/3-3$. We first give a completely elementary proof that $g(r)\\ge 3r-4$. We then build on the same approach and apply Kahn's small-codegree hypergraph edge-colouring theorem to improve this to $g(r)\\ge (61/20-o(1))r$. To the best of our knowledge, this is the first improvement over the Erd\\H{o}s-Lov\\'asz lower bound in about fifty years."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24878","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/2606.24878/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"}