{"paper":{"title":"Set-polynomials and polynomial extension of the Hales-Jewett Theorem","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Leibman, Vitaly Bergelson","submitted_at":"1999-07-01T00:00:00Z","abstract_excerpt":"An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established:\n  Theorem. Let r,d,q \\in \\N. There exists N \\in \\N such that for any r-coloring of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a \\subset V and a nonempty set \\gamma \\subseteq {1,...,N} such that a \\cap (\\gamma^{d} x {1,...,q}) = \\emptyset, and the subsets a, a \\cup (\\gamma^{d} x {1}), a \\cup (\\gamma^{d} x {2}), ..., a \\cup (\\gamma^{d} x {q}) are all of the same color.\n  This ``polynomial'' Hales-Jewett theorem contains refinements of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9907201","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"}