Uniformly Dense Matroid

A uniformly dense matroid is a finite matroid on ground set whose matroid rank function satisfies, for every subset ,

For , this is equivalent to , so no subset has a larger ratio of cardinality to matroid rank than the whole ground set (Kajitani et al. 1988).

Every matroid having a cyclic basis ordering satisfies this inequality. The Kajitani-Ueno-Miyano conjecture asserts the converse.