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2000

Simple Matroids With Bounded Cocircuit Size

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Simple Matroids With Bounded Cocircuit Size
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 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.
Joseph E. Bonin, Talmage James Reid
Added 17 Dec 2010
Updated 17 Dec 2010
Type Journal
Year 2000
Where CPC
Authors Joseph E. Bonin, Talmage James Reid
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