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On the asymptotic proportion of connected matroids. (English) Zbl 1244.05047
Random graphs and their properties are well studied, but very little is known about the asymptotic behavior of classes of matroids. The proportion of graphs on \(n\) vertices with non-trivial automorphism group tends to zero as \(n\) tends to infinity and this is true for labeled as well as unlabeled graphs. The authors conjecture that asymptotically almost every matroid has trivial automorphism group, is arbitrarily highly connected and is not representable over any field. A proof is provided for the fact that the proportion of \(n\)-labeled matroids that are connected is asymptotically at least \(1/2\). The proof even of this most likely not best possible bound is highly non-trivial and suggests that the conjectures formulated in the paper are challenging.

05B35 Combinatorial aspects of matroids and geometric lattices
05A16 Asymptotic enumeration
05C80 Random graphs (graph-theoretic aspects)
Full Text: DOI
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