Matroids with no \((q+2)\)-point-line minors (Q675880)

This paper is concerned with the interplay between the number of points in a matroid and the structural and representability properties of the matroid. The paper is entirely devoted to the class \({\mathcal U}(q)\) of matroids with no \((q+2)\)-point-line minor and subclasses. The paper starts by giving an alternative proof of a result by \textit{J. P. S. Kung} [Contemp. Math. 147, 21-61 (1993; Zbl 0791.05018)] that says that a rank-\(r\) matroid in \({\mathcal U}(q)\) has at most \((q^r-1)/(q-1)\) points. Then an ``affine'' version of the result is proved that says that a rank-\(r\) matroid in \({\mathcal U}(q)\) with no \((q+1)\)-point-line minor has at most \(q^{r-1}\) points and this bound is assumed only by affine geometries. Then a result concerning representability of matroids is given. For \(r > 3\) and \(q\) a prime power, any matroid in \({\mathcal U}(q)\) with at least \(q^{r-1}\) points is representable over the field with \(q\) elements. If \(r > 3\) and \(q\) is not a prime power then any rank-\(r\) matroid in \({\mathcal U}(q)\) has at least \(q^{r-1}-1\) points. Some more results in this direction are proved. Then a new proof of the classical result by Tutte that \({\mathcal U}(2)\) is exactly the class of matroids representable over the field with \(2\) elements is presented. The paper closes with results on unique representability.