Critical Exponents, Colines, and Projective Geometries

In [9, p. 469], Oxley made the following conjecture, which is a geometric analogue of a conjecture of Lovasz (see [1, p. 290]) about complete graphs.Conjecture 1.1. Let G be a rank-n GF(q)-representable simple matroid with critical exponent n − γ. If, for every coline X in G, c(G/X; q) = c(G; q) − 2 = n − γ − 2, then G is the projective geometry PG(n − 1, q).We shall call the rank n, the critical ‘co-exponent’ γ, and the order q of the field the parameters of Oxley's conjecture. We exhibit several counterexamples to this conjecture. These examples show that, for a given prime power q and a given positive integer γ, Oxley's conjecture holds for only finitely many ranks n. We shall assume familiarity with matroid theory and, in particular, the theory of critical problems. See [6] and [9].A subset C of points of PG(n − 1, q) is a (γ, k)-cordon if, for every k-codimensional subspace X in PG(n − 1, q), the intersection C ∩ X contains a γ-dimensional subspace of PG(n − 1, q). In this paper, our primary interest will be in constructing (γ, 2)-cordons. With straightforward modifications, our methods will also yield (γ, k)-cordons.Complements of counterexamples to Oxley's conjecture are (γ, 2)-cordons.