{"paper":{"title":"A density version of the Carlson--Simpson theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konstantinos Tyros, Pandelis Dodos, Vassilis Kanellopoulos","submitted_at":"2012-09-22T11:34:41Z","abstract_excerpt":"We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following.\n  For every integer $k\\geq 2$ and every set $A$ of words over $k$ satisfying \\[\\limsup_{n\\to\\infty} \\frac{|A\\cap [k]^n|}{k^n}>0\\] there exist a word $c$ over $k$ and a sequence $(w_n)$ of left variable words over $k$ such that the set \\[\\{c\\}\\cup \\big\\{c^{\\smallfrown}w_0(a_0)^{\\smallfrown}...^{\\smallfrown}w_n(a_n) : n\\in\\mathbb{N} \\ \\text{ and } \\ a_0,...,a_n\\in [k]\\big\\}\\] is contained in $A$.\n  While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative ve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4985","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":""},"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"}