Odd circuits in dense binary matroids

Abstract

The exclusion of odd circuits from a binary matroid here is natural. The geometric density Hales-Jewett theorem [3] implies that dense GF(q)representable matroids with sufficiently large rank necessarily contain arbitrarily large affine geometries over GF(q); these geometries contain circuits of every possible even cardinality when q= 2 and circuits of every possible cardinality when q>2. So dense k-circuit free GF(q)-representable matroids of large rank only exist when q=2 and k is odd. Our main theorem (Theorem 3.1) is somewhat more general than Theorem 1.1; for each odd k≥5, it bounds the critical number of all sufficiently

DOI: 10.1007/s00493-015-3237-1

Cite this paper

@article{Geelen2017OddCI, title={Odd circuits in dense binary matroids}, author={James F. Geelen and Peter Nelson}, journal={Combinatorica}, year={2017}, volume={37}, pages={41-47} }