{"paper":{"title":"Phase transition of degenerate Tur\\'{a}n problems in $p$-norms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Jun Gao, Oleg Pikhurko, Xizhi Liu","submitted_at":"2024-11-23T14:56:35Z","abstract_excerpt":"For a positive real number $p$, the $p$-norm $\\left\\lVert G \\right\\rVert_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices. F\\\"{u}redi and K\\\"{u}ndgen \\cite{FK06} show that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions, assuming a mild assumption on the growth rate of $\\mathrm{ex}(n,F)$. The main contr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15579","kind":"arxiv","version":3},"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/2411.15579/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"}