A counterexample to Aharoni's strongly maximal matching conjecture

It is conjectured (and proved for edge sets of graphs) in [1] that in every family d of finite sets a subfamily ~ of disjoint sets (called a 'strongly maximal matching') exists, so that no replacement of k of them by more than k sets from a / resu l t s again in a subfamily of disjoint sets. As expected by Erd6s (Introduction of [2]), the conjecture is false. A counterexample is ag, the family of those finite subsets of the set N of natural numbers, whose cardinality and smallest element (in canonical order) are equal. In fact, suppose ,~/ contains a strongly maximal matching ~, then, by our definitions ~ is infinite, has an element B = {bl < b2 < "" < bt} with b I --t >1 3 and