{"paper":{"title":"Lagrangians of hypergraphs: The Frankl-F\\\"uredi conjecture holds almost everywhere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.CO","authors_text":"Mykhaylo Tyomkyn","submitted_at":"2017-03-13T07:25:58Z","abstract_excerpt":"Frankl and F\\\"uredi conjectured in 1989 that the maximum Lagrangian of all $r$-uniform hypergraphs of fixed size $m$ is realised by the initial segment of the colexicographic order. In particular, in the principal case $m=\\binom{t}{r}$ their conjecture states that every $H\\subseteq \\mathbb{N}^{(r)}$ of size $\\binom{t}{r}$ satisfies \\begin{align*} \\max \\{\\sum_{A \\in H}\\prod_{i\\in A} y_i \\ \\colon \\ y_1,y_2,\\ldots \\geq 0; \\sum_{i\\in \\mathbb{N}} y_i=1 \\}&\\leq \\frac{1}{t^r}\\binom{t}{r}. \\end{align*}\n  We prove the above statement for all $r\\geq 4$ and large values of $t$ (the case $r=3$ was settled"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.04273","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"}