Abstract
A hereditary class ℋ of combinatorial geometries (or simple matroids) is a collection of geometries closed under minors and direct sums. A geometry G in ℋ is extremal if no proper extension of G of the same rank is in ℋ. The size function h(n) of ℋ is defined by h(n)=max {|G|: G∈ℋ and rank(G)=n}, where |G| is the number of points in G. A hereditary class is numerically regular if for every extremal geometry G in ℋ, |G|=h (rank(G)). We determine all the numerically regular hereditary classes for which the set {h(n)−h(n−1): 1≤n<∞} of positive integers does not have an upper bound: they are all varieties. We also give several examples of numerically regular hereditary classes which are not varieties.
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Partially supported by a North Texas State University Faculty Research Grant.
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Kung, J.P.S. Numerically regular hereditary classes of combinatorial geometries. Geom Dedicata 21, 85–105 (1986). https://doi.org/10.1007/BF00147534
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DOI: https://doi.org/10.1007/BF00147534