On Infinite Antichains of Matroids

Robertson and Seymour have shown that there is no infinite set of graphs in which no member is a minor of another. By contrast, it is well known that the class of all matroids does contains such infinite antichains. However, for many classes of matroids, even the class of binary matroids, it is not known whether or not the class contains an infinite antichain. In this paper, we examine a class of matroids of relatively simple structure: Ma,b,c consists of those matroids for which the deletion of some set of at most a elements and the contraction of some set of at most b elements results in a matroid in which every component has at most c elements. We determine precisely when Ma,b,c contains an infinite antichain. We also show that, among the matroids representable over a finite fixed field, there is no infinite antichain in a fixed Ma,b,c; nor is there an infinite antichain when the circuit size is bounded.