MATROIDS WITH NO (q + 2)-POINT-LINE MINORS

It is known that a geometry with rank r and no minor isomorphic to the (q + 2)-point line has at most (qr − 1)/(q − 1) points, with strictly fewer points if r > 3 and q is not a prime power. For q not a prime power and r > 3, we show that qr−1 − 1 is an upper bound. For q a prime power and r > 3, we show that any rank-r geometry with at least qr−1 points and no (q + 2)-point-line minor is representable over G(q). We strengthen these bounds to qr−1− (qr−2−1)/(q−1)−1 and qr−1− (qr−2−1)/(q−1) respectively when q is odd. We give an application to unique representability and a new proof of Tutte’s theorem: A matroid is binary if and only if the 4-point line is not a minor. 1. SOME EXTREMAL MATROIDS: PROJECTIVE AND AFFINE GEOMETRIES We are concerned with matroids containing no minor isomorphic to the (q + 2)-point line, i.e., the uniform matroid U2,q+2. These matroids form a minor-closed class, denoted U(q). This class is of interest in part because of its connections with representability questions and its role in extremal matroid theory, in particular, in connection with size functions (see Section 4.2 of [6], especially Corollary 4.5). If q is a prime power, then L(q) ⊆ U(q), where L(q) is the class of matroids representable over G(q). Tutte [10] proved that L(2) = U(2). The containment is strict for all other prime powers q. The starting point for our work is the following result [6, Theorem 4.3]. Theorem 1. Rank-r geometries in U(q) have at most (qr−1)/(q−1) points. This upper bound is attained only by projective geometries of order q. Thus for r > 3, this bound is attained if and only if q is a prime power. To set the stage for the rest of the paper, we start by giving an alternate proof of Theorem 1, using the axioms of projective geometry. To prove the upper bound in Theorem 1, assume M is a rank-r geometry in U(q) with n points. Consider M/x, the contraction of M by the point x. Since lines through x contain at most q points in addition to x, the simplification of M/x has at least (n − 1)/q points. Thus if M has more than (q − 1)/(q− 1) points, the simplification of M/x has more than (qr−1− 1)/(q− 1) points. Contracting r − 2 times yields a line with at least q + 2 points, contrary to the assumption. The next observation follows immediately from these ideas and is useful for analyzing the case in which the upper bound is attained. Lemma 1. For any rank-r geometry M in U(q) with (q − 1)/(q− 1) points, the lines through any point x partition the points of M − x into blocks of size q. It follows from Lemma 1 that for any rank-r geometry M in U(q) with (q − 1)/(q − 1) points, each line contains exactly q+ 1 points. Furthermore, if r = 3 each point is on exactly q+ 1 lines. From this, it is easy to check that all (q + q + 1)-point planes with no (q + 2)-point-line minor are projective planes of order q. Recall that a geometry is a projective geometry if and only if any two coplanar lines intersect in a point and all lines have three or more points. Thus to prove that rank-r geometries in U(q) with (qr−1)/(q−1) points are projective geometries, it suffices to prove that coplanar lines intersect and that each line has q+1 points. We have already verified the latter condition. The next lemma shows that every hyperplane of M