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Abstract

Assign independent priorities from one common atomless law to the elements of a finite matroid and construct its minimum-weight basis greedily. We bound every joint inclusion probability in terms of the corresponding individual marginals, with a factorial coefficient and an explicit marginal-dependent improvement. The pair case gives a universal factor two, improving the previously stated factor eight for random minimum spanning trees. The proof first conditions on all unmarked priorities. Forcing a marked element into the basis and contracting it yields a conditional cylinder inequality; conditional Jensen and Hölder inequalities then return to the original one-element closure thresholds. Bernoulli sprinkling bounds these thresholds in convex order by a beta distribution with the same mean. We obtain a sharp moment hierarchy, characterize equality by rank-one direct summands, and derive exclusion and occupancy bounds. Finite-cylinder limits transfer the estimates to countable finitary-matroid restriction limits and to both free and wired minimal spanning forests. On transitive unimodular graphs this gives a uniform exponential bound for the wired-forest degree distribution.