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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.

Most cited references6

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Regularity and positional games

(1963)
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Growth rates of minor-closed classes of matroids

(2009)
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Cliques in dense GF(q)-representable matroids

(2003)
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Author and article information

Journal
16 October 2012
Article
1210.4522