{"paper":{"title":"A Tur\\'an Theorem for Cayley Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kai Yang, Wei Li","submitted_at":"2026-06-28T09:08:40Z","abstract_excerpt":"In this note, we give a Tur\\'an theorem for Cayley graphs $\\Cay(\\Z_p,S)$ over prime cyclic groups $\\Z_p$. For a graph $F$ and a finite abelian group $G$, define the Cayley--Tur\\'an number by \\[ \\exCay(F,G) =\n\\max\\{|S|:S=-S\\subseteq G\\setminus\\{0\\},\\ \\Cay(G,S)\\text{ is }F\\text{-free}\\}. \\] Using a polynomial method, we prove that for every odd prime $p$ and every $1\\le r\\le p-1$, \\[ \\exCay(K_{r+1},\\Z_p) =\np-1-2\\left\\lfloor\\frac{p}{r+1}\\right\\rfloor . \\] The extremal construction is the complement of the short-difference interval \\[ D_0=\\{0,\\pm1,\\ldots,\\pm\\lfloor p/(r+1)\\rfloor\\}. \\] We also dis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29284","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/2606.29284/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"}