{"paper":{"title":"A Simple Counting Argument for Dense Linear Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'ozsef Solymosi, Lior Gishboliner","submitted_at":"2026-06-24T15:07:33Z","abstract_excerpt":"In connection to the Brown-Erd\\H{o}s-S\\'os conjecture, we give a short local averaging proof of a density theorem for linear uniform hypergraphs. Let $r \\ge 3$, $k \\ge 3$, and suppose that $n \\ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and \\[|E(H)| \\geq \\frac{k-2}{r^2((r-2)(k-2)+1)}n^2 + \\frac{n}{r},\\] then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c \\geq \\frac{r-1}{r} \\cdot \\frac{k-2}{(r-2)(k-2)+1} + o(1)$. In particular, this yields a simple proof of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25931","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/2606.25931/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"}