## 1 0 O ct 2 01 7 STABILITY AND EXACT TURÁN NUMBERS FOR MATROIDS

- HONG LIU, SAMMY LUO, PETER NELSON, KAZUHIRO NOMOTO
- 2017

1 Excerpt

- Published 2017 in Combinatorica

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

@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}
}