Matroids are Equitable

We show that if the ground set of a matroid can be partitioned into $k\ge 2$ bases, then for any given subset $S$ of the ground set, there is a partition into $k$ bases such that the sizes of the intersections of the bases with $S$ may differ by at most one. This settles the matroid equitability conjecture by Fekete and Szab\'o (Electron. J. Comb. 2011) in the affirmative. We also investigate equitable splittings of two disjoint sets $S_1$ and $S_2$, and show that there is a partition into $k$ bases such that the sizes of the intersections with $S_1$ may differ by at most one and the sizes of the intersections with $S_2$ may differ by at most two; this is the best one can hope for arbitrary matroids. We also derive applications of this result into matroid constrained fair division problems. We show that there exists a matroid-constrained fair division that is envy-free up to one item if the valuations are identical and tri-valued additive. We also show that for bi-valued additive valuations, there exists a matroid-constrained allocation that provides everyone their maximin share.