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For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that $$\Omega \left( {1/n} \right) \leqslant 1 - b\left( M \right) \leqslant O\left( {\log {{\left( n \right)}^3}/n} \right)a\;sn \to \infty $$ Ω ( 1 / n ) ≤ 1 − b ( M ) ≤ O ( log ( n ) 3 / n ) a s n → ∞ for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U k,2k -minor, whenever k≤O(log(n)), (2) have girth ≥Ω(log(n)), (3) have Tutte connectivity $$\geq\Omega\;{(\sqrt {log(n)})}$$ ≥ Ω ( l o g ( n ) ) , and (4) do not arise as the truncation of another matroid.
Combinatorica – Springer Journals
Published: Jun 5, 2018
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