A generalisation of uniform matroids

A matroid is uniform if and only if it has no minor isomorphic to U1,1⊕U0,1 and is paving if and only if it has no minor isomorphic to U2,2⊕U0,1. This paper considers, more generally, when a matroid M has no Uk,k⊕U0,ℓ-minor for a fixed pair of positive integers (k,ℓ). Calling such a matroid (k,ℓ)-uniform, it is shown that this is equivalent to the condition that every rank-(r(M)−k) flat of M has nullity less than ℓ. Generalising a result of Rajpal, we prove that for any pair (k,ℓ) of positive integers and prime power q, only finitely many simple cosimple GF(q)-representable matroids are (k,ℓ)-uniform. Consequently, if Rota's Conjecture holds, then for every prime power q, there exists a pair (kq,ℓq) of positive integers such that every excluded minor of GF(q)-representability is (kq,ℓq)-uniform. We also determine all binary (2,2)-uniform matroids and show the maximally 3-connected members to be Z5﹨t,AG(4,2),AG(4,2)⁎ and a particular self-dual matroid P10. Combined with results of Acketa and Rajpal, this completes the list of binary (k,ℓ)-uniform matroids for which k+ℓ≤4.