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,..., kn which contains no combinatorial line; similarly, the Moser number c nk is the largest subset of the cube [k]n which contains no geometric line. A deep theorem of Furstenberg and Katznelson , ,  shows that cn,k=o(kn) as n→∞ (which implies a similar claim for cn); this is already non-trivial for k = 3. Several new proofs of this result have also been recently established , .
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