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Extremal matroid theory. (English) Zbl 0791.05018
Robertson, Neil (ed.) et al., Graph structure theory. Proceedings of the AMS-IMS-SIAM joint summer research conference on graph minors held June 22 to July 5, 1991 at the University of Washington, Seattle, WA (USA). Providence, RI: American Mathematical Society. Contemp. Math. 147, 21-61 (1993).
As matroid theory is the common generalization of graph theory and projective geometry, one can, in an expansionist mood, classify every extremal problem in either area under extremal matroid theory. But, more plausibly, the reverse is the case: extremal matroid theory is motivated by and derives many—but not all—of its problems and methods from graphical and geometric extremal problems. In this survey, we shall begin by discussing several classical extremal theorems and problems connected with matroids. Next, in §3, we shall present results about excluding submatroids. In §§4, 5, and 6, results about excluding minors will be discussed. Finally, in §7, we shall discuss the matroid version of the direction problem in real and complex space. It goes without saying that any survey reflects the philosophy and resarch work of its author. This survey concentrates on what I personally find interesting.
For the entire collection see [Zbl 0777.00050].

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05C35 Extremal problems in graph theory
05C99 Graph theory
06C10 Semimodular lattices, geometric lattices
51M04 Elementary problems in Euclidean geometries
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
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