{"paper":{"title":"New constructions and bounds for nonabelian Sidon sets with applications to Tur\\'an-type problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"John Byrne, Michael Tait","submitted_at":"2025-09-09T13:51:56Z","abstract_excerpt":"An $S_k$-set in a group $\\Gamma$ is a set $A\\subseteq\\Gamma$ such that $\\alpha_1\\cdots\\alpha_k=\\beta_1\\cdots\\beta_k$ with $\\alpha_i,\\beta_i\\in A$ implies $(\\alpha_1,\\ldots,\\alpha_k)=(\\beta_1,\\ldots,\\beta_k)$. An $S_k'$-set is a set such that $\\alpha_1\\beta_1^{-1}\\cdots\\alpha_k\\beta_k^{-1}=1$ implies that there exists $i$ such that $\\alpha_i=\\beta_i\\text{ or }\\beta_i=\\alpha_{i+1}$. We give explicit constructions of large $S_k$-sets in the group $S_n$ and $S_2$-sets in $S_n\\times S_n$ and $A_n\\times A_n$. We give probabilistic constructions for `nice' groups which obtain large $S_2$-sets in $A_n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.07750","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/2509.07750/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"}