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    Projective geometries in exponentially dense matroids. II

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    We show for each positive integer aa that, if M\mathcal{M} is a minor-closed class of matroids not containing all rank-(a+1)(a+1) uniform matroids, then there exists an integer cc such that either every rank-rr matroid in M\mathcal{M} can be covered by at most rcr^c rank-aa sets, or M\mathcal{M} contains the GF(q)(q)-representable matroids for some prime power qq and every rank-rr matroid in M\mathcal{M} can be covered by at most cqrcq^r rank-aa sets. In the latter case, this determines the maximum density of matroids in M\mathcal{M} up to a constant factor

    Projective geometries in exponentially dense matroids. I

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    We show for each positive integer aa that, if \cM is a minor-closed class of matroids not containing all rank-(a+1)(a+1) uniform matroids, then there exists an integer nn such that either every rank-rr matroid in \cM can be covered by at most rnr^n sets of rank at most aa, or \cM contains the \GF(q)-representable matroids for some prime power qq, and every rank-rr matroid in \cM can be covered by at most rnqrr^nq^r sets of rank at most aa. This determines the maximum density of the matroids in \cM up to a polynomial factor
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