A combinatorial structure on a ground set E consisting of a family I of independent subsets of E satisfying (o) the empty set is independent (i) a subset of an independent set is independent (ii) if X and Y are independent and |X| > |Y| then there is an x in X such that Y union {x} is also independent.

Examples:
(1) independent sets in a finite-dimensional vector space
(2) subsets of the vertex set of a graph containing no circuit
(3) the uniform matroid all subsets of E of size not greater than a fixed r

Balanced Incomplete Block Design.

A square array of size n each containing a smybol from an alphabet of size n in such a way that each smybol occurs exactly once in each row and column.