The Rédei Function of a Relation

Abstract The critical problem of matroid theory can be posed in the more general context of finite relations. Given a relation R between the finite sets S and T, the critical problem is to determine the smallest number n such that there exists an n-tuple (u1,…, un) of elements from T such that for every x in S, there exists a u1 such that xRui. All the enumerative results, in particular, the Tutte decomposition and Mobius function formula, can be rephrased so that they still hold. In this way, we obtain a uniform approach to all the classical critical problems.