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      A Density Hales-Jewett Theorem for matroids

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          Abstract

          We show that, if \(\alpha > 0\) is a real number, \(n \ge 2\) and \(\ell \ge 2\) are integers, and \(q\) is a prime power, then every simple matroid \(M\) of sufficiently large rank, with no \(U_{2,\ell}\)-minor, no rank-\(n\) projective geometry minor over a larger field than \(\GF(q)\), and satisfying \(|M| \ge \alpha q^{r(M)}\), has a rank-\(n\) affine geometry restriction over \(\GF(q)\). This result can be viewed as an analogue of the Multidimensional Density Hales-Jewett Theorem for matroids.

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          Most cited references 6

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          Regularity and positional games

           A. Hales,  R. Jewett (1963)
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            Growth rates of minor-closed classes of matroids

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              Cliques in dense GF(q)-representable matroids

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                Author and article information

                Journal
                16 October 2012
                Article
                1210.4522

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math.CO

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