Simple Matroids with Bounded Cocircuit Size

  • Authors:

  • Joseph E. Bonin;Talmage James Reid

  • Affiliations:

  • Department of Mathematics, The George Washington University, Washington, DC 20052, USA (e-mail: jbonin@gwis2.circ.gwu.edu);Department of Mathematics, The University of Mississippi, University, MS 38677, USA (e-mail: mmreid@hilbert.math.olemiss.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2000

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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⌋ points. We show that if M is a connected matroid of rank 4 and d is q3 with q 1, then M has at most q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case the only such matroid with exactly q3 + q2 + q + 1 points is the projective geometry PG(3, q). We also show that if d is q4 for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most q4 + q3 + q2 + 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.