ON INFINITE ANTICHAINS OF MATROIDS

Robertson and Seymour have shown that there is no infinite set of grap hs in which no member is a minor of another. By contrast, it is well k nown that the class of all matroids does contains such infinite antich ains. However, for many classes of matroids, even the class of binary matroids, it is not known whether or not the class contains an infinit e antichain. In this paper, we examine a class of matroids of relative ly simple structure: M(a,b,c) consists of those matroids for which the deletion of some set of at most a elements and the contraction of som e set of at most b elements results in a matroid in which every compon ent has at most c elements. We determine precisely when M(a,b,c) conta ins an infinite antichain. We also show that, among the matroids repre sentable over a finite fixed field, there is no infinite antichain in a fixed M(a,b,c); nor is there an infinite antichain when the circuit size is bounded. (C) 1995 Academic Press, Inc.