{"paper":{"title":"The Maximum Number of Three Term Arithmetic Progressions, and Triangles in Cayley Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zachary Chase","submitted_at":"2018-09-11T08:21:24Z","abstract_excerpt":"Let $G$ be a finite Abelian group. For a subset $S \\subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \\frac{1}{|S|^2}\\sum_{x,y \\in S} 1_S(x+y)$. For any $q \\ge 1$ and $\\alpha \\in [0,1]$, and any $S \\subseteq G$ with $|S| = \\frac{|G|}{q+\\alpha}$, we show $\\frac{T_3(S)}{|S|^2}$ and Prob[$S$] are bounded above by $\\max\\left(\\frac{q^2-\\alpha q+\\alpha^2}{q^2},\\frac{q^2+2\\alpha q+4\\alpha^2-6\\alpha+3}{(q+1)^2},\\gamma_0\\right)$, where $\\gamma_0 < 1$ is an absolute constant. As a consequence, we verify a graph theoretic conjecture of Gan, Loh, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.03729","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"}