Discrete Mathematics 270 (2003) 225 – 226 www.elsevier.com/locate/disc Erratum Erratum to “On a class of nite linear spaces with few lines” [Discrete Mathematics 270 (2003) 207–224] Vito Napolitano Dipartimento di Matematica, Universita degli Studi Della Basilicata, Campus Macchia Romana, Contrada Macchia Romana, 85100 Potenza, Italy The statement of Theorem I in the paper is not correct, in fact there are some missing cases. Clearly, the true statement is the following. Theorem I. Let (P; L) be a nite linear space on v points, such that b − v 6 m. Then one of the following holds. (1) (P; L) is the near pencil on v points, the (3,3)-cross, the (3,4)-cross or the linear space on v = 5 points, with a line of length 3 and all the other of length 2. (2) (P; L) is an ane plane of order m − 1 or a punctured ane plane of order m − 1. (3) (P; L) is an ane plane of order m with a point at in nity. (4) (P; L) is a punctured ane plane of order m with a point at in nity. (5) (P; L) is obtained from a nite projective plane of order m − 1 by deleting at most m points. (6) (P; L) is an ane plane of order m−1 with either a punctured projective plane, or the (3,3)-cross at in nity. (7) (P; L) is a projectively in ated punctured ane plane of order m − 1, with at least four points at in nity. (8) A projectively in ated ane plane of order m − 1. (9) (P; L) is a punctured ane plane of order m − 1 with a triangle at in nity. (10) (P; L) is the linear space on v = 7 points, b = 10 lines, m = 3, with a single line of length k = m + 1 = 4, three lines of length 3 and the remaining lines of length 2.  doi of the original article: 10.1016/S0012-365X(02)00873-7 E-mail address: vnapolitano@unibas.it (V. Napolitano). c 2003 Elsevier B.V. All rights reserved. 0012-365X/03/$ - see front matter
doi:10.1016/S0012-365X(03)00112-2 226 V. Napolitano / Discrete Mathematics 270 (2003) 225 – 226 (11) (P; L) is the linear space on v = 8 points, b = 11 lines, m = 3, with a single line L of length k = m + 1 = 4, six lines of length 3 and the remaining lines of length 2, and on each point of L there is a line of length 3. (12) (P; L) is the linear space on v = 8 points, b = 11 lines, m = 3, with a single line L of length k = m + 1 = 4, six lines of length 3 and the remaining lines of length 2, and with a point L on which there is no line of length 3. Furthermore, in the proof of Theorem I in the paper (at the beginning of Section 2.2.2 case k = m), the statement: by Theorem B we may assume that the maximum point degree is m + 1, is incomplete. Actually, there is a missing proposition. The correct statement is as follows. By Theorem B and the following proposition, one may assume that the maximum point degree is m + 1. Proposition. A nite linear space (P; L) with b − v = k = m, and with a point of degree at least m + 2 is either an ane plane of order m − 1 with a punctured projective plane or the (3,3)-cross at in nity, or a projectively in ated ane plane of order m − 1, with at least four points at in nity. Proof. By Theorem B in the paper (P; L) is an s-fold in ated ane plane of order m − 1. Let v and b be the number of points and lines of (P; L), respectively. Then s s

b = (m − 1)2 + m − s + bi and v = (m − 1)2 + m − d − (m − vi ); i=1 i=1 where vi and bi are the number of points and lines of the ith main subspace of (P; L), and d is the de ciency of (P; L). Hence
m=b−v= (bi − vi ) + s(m − 1) + d: It follows that m ¿ s(m − 1), and so s 6 2. Since m ¿ 3 one has s = 1. Thus m = b1 − v1 + m − 1 + d; which implies d = 0 and b1 − v1 = 1, or d = 1 and b1 − v1 = 0, and so (P; L) is one of the linear spaces described in the assertion.