{"paper":{"title":"Subsets of Products of Finite Sets of Positive Upper Density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konstantinos Tyros, Stevo Todorcevic","submitted_at":"2012-11-16T16:38:22Z","abstract_excerpt":"In this note we prove that for every sequence $(m_q)_{q}$ of positive integers and for every real $0<\\delta\\leqslant1$ there is a sequence $(n_q)_{q}$ of positive integers such that for every sequence $(H_q)_{q}$ of finite sets such that $|H_q|=n_q$ for every $q\\in\\mathbb{N}$ and for every $D\\subseteq \\bigcup_k\\prod_{q=0}^{k-1}H_q$ with the property that $$\\limsup_k \\frac{|D\\cap \\prod_{q=0}^{k-1} H_q|}{|\\prod_{q=0}^{k-1}H_q|}\\geqslant\\delta$$ there is a sequence $(J_q)_{q}$, where $J_q\\subseteq H_q$ and $|J_q|=m_q$ for all $q$, such that $\\prod_{q=0}^{k-1}J_q\\subseteq D$ for infinitely many $k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3948","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"}