The structure of matroids with a spanning clique or projective geometry.

Let s , n ≥ 2 be integers. We give a qualitative structural description of every matroid M that is spanned by a frame matroid of a complete graph and has no U s , 2 s -minor and no rank- n projective geometry minor, showing that every such matroid is ‘close’ to a frame matroid. We also give a similar description of every matroid M with a spanning projective geometry over a field GF ( q ) as a restriction and with no U s , 2 s -minor and no PG ( n , q ′ ) -minor for any q ′ > q , showing that such an M is ‘close’ to a GF ( q ) -representable matroid. [ABSTRACT FROM AUTHOR]