{"paper":{"title":"Triple systems with no three triples spanning at most five points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Stefan Glock","submitted_at":"2018-09-06T17:16:52Z","abstract_excerpt":"We show that the maximum number of triples on $n$~points, if no three triples span at most five points, is $(1\\pm o(1))n^2/5$. More generally, let $f^{(r)}(n;k,s)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$~vertices not containing a subgraph with $k$~vertices and $s$~edges. In 1973, Brown, Erd\\H{o}s and S\\'os conjectured that the limit $\\lim_{n\\to \\infty}n^{-2}f^{(3)}(n;k,k-2)$ exists for all~$k$. They proved this for $k=4$, where the limit is $1/6$ and the extremal examples are Steiner triple systems. We prove the conjecture for $k=5$ and show that the limit is~$1/5$. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02100","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":""},"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"}