Simple Matroids with Bounded Cocircuit Size

Abstract

We examine the specialization to simple matroids of certain problems in extremal matroid theory that are concerned with bounded cocircuit size. Assume that each cocircuit of a simple matroid M has at most d elements. We show that if M has rank 3, then M has at most d + ⌊ √ d⌋ + 1 points and we classify the rank-3 simple matroids M that have exactly d + ⌊ √ d⌋ + 1 points. We show that if M is a connected matroid of rank 4 and d is q 3 with q > 1, then M has at most q 3 + q 2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case the only such matroid with exactly q 3 + q 2 + q + 1 points is the projective geometry PG(3, q). We also show that if d is q 4 for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most q 4 + q 3 + q 2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case PG(4, q) is the only such matroid that attains this bound.