{"paper":{"title":"High-order Phase Transition in Random Hypergrpahs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Xing Peng","submitted_at":"2014-09-03T17:23:13Z","abstract_excerpt":"In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set is selected as an edge with probability $p$ independently randomly. For $1\\leq s \\leq r-1$ and two $s$-sets $S$ and $S'$, we say $S$ is connected to $S'$ if there is a sequence of alternating $s$-sets and edges $S_0,F_1,S_1,F_2, \\ldots, F_k, S_k$ such that $S_0,S_1,\\ldots, S_k$ are $s$-sets, $S_0=S$, $S_k=S'$, $F_1,F_2,\\ldots, F_k$ are edges of $H$, and $S_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1174","kind":"arxiv","version":2},"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"}