{"paper":{"title":"Effectiveness of Hindman's theorem for bounded sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Carl G. Jockusch, Damir D. Dzhafarov, Jr., Linda Brown Westrick, Reed Solomon","submitted_at":"2016-03-27T19:45:01Z","abstract_excerpt":"We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\\mathsf{HT}^{\\leq n}_k$ denote the assertion that for each $k$-coloring $c$ of $\\mathbb{N}$ there is an infinite set $X \\subseteq \\mathbb{N}$ such that all sums $\\sum_{x \\in F} x$ for $F \\subseteq X$ and $0 < |F| \\leq n$ have the same color. We prove that there is a computable $2$-coloring $c$ of $\\mathbb{N}$ such that there is no infinite computable set $X$ such that all nonempty sums of at most $2$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08249","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"}