The number of lines in a matroid with no $U_{2,n}$-minor

TL;DR

This paper establishes bounds on the number of lines in certain matroids without specific minors, showing the maximum matches projective geometries over GF(q) for small q, but not for larger q.

Contribution

It proves that for prime powers up to 5, the maximum lines in a matroid without a U_{2,q+2} minor equals that of a projective geometry over GF(q), and provides counterexamples for larger q.

Findings

Maximum lines match projective geometries over GF(q) for q ≤ 5.

Counterexamples exist for larger prime powers.

Bound does not hold universally for all prime powers.

Abstract

We show that, if $q$ is a prime power at most 5, then every rank-$r$ matroid with no $U_{2,q+2}$-minor has no more lines than a rank-$r$ projective geometry over GF$(q)$. We also give examples showing that for every other prime power this bound does not hold.