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DENSITY HALES-JEWETT AND MOSER NUMBERS IN LOW DIMENSIONS

By D. H. J. Polymath

Abstract

Abstract. For any n ≥ 0 and k ≥ 1, the density Hales-Jewett number cn,k is defined as the size of the largest subset of the cube [k] n: = {1,..., k} n which contains no combinatorial line; similarly, the Moser number c ′ n,k is the largest subset of the cube [k] n which contains no geometric line. A deep theorem of Furstenberg and Katznelson [7], [8], [13] shows that cn,k = o(kn) as n → ∞ (which implies a similar claim for c ′ n,k; this is already non-trivial for k = 3. Several new proofs of this result have also been recently established [16], [1]. Using both human and computer-assisted arguments, we compute several values of cn,k and c ′ n, k for small n, k. For instance the sequence cn,3 for n = 0,..., 6 i

Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.194.2204
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