Binary matroids with no \(PG(3,2)\)-minor. (English) Zbl 1160.05012
Summary: We show that a binary rank-\(r\) matroid with more than \(15r(r+1)/4\) points contains a minor isomorphic to the rank-4 binary projective geometry \(PG(3,2)\).
MSC:
05B35 | Combinatorial aspects of matroids and geometric lattices |
05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
05C35 | Extremal problems in graph theory |
05D99 | Extremal combinatorics |
06C10 | Semimodular lattices, geometric lattices |
51M04 | Elementary problems in Euclidean geometries |