Linear and Multilinear Algebra, Vol. 53, No. 6, 2005, 435–443 Possible numbers of ones in 0–1 matrices with a given rank QI HU, YAQIN LI and XINGZHI ZHAN* Department of Mathematics, East China Normal University, Shanghai 200062, China Communicated by R.B. Bapat (Received 15 October 2004) We determine the possible numbers of ones in a 0–1 matrix with given rank in the generic case and in the symmetric case. There are some unexpected phenomena. The rank 2 symmetric case is subtle. Keywords: 0–1 Matrix; Rank; Number of ones; Symmetric matrix AMS Classifications: 15A03; 15A36; 05B20 1. Introduction A 0–1 matrix is a matrix whose entries are either 0 or 1. Such matrices arise frequently in combinatorics and graph theory. It is known [1, p. 243] that the largest number of ones in an n  n nonsingular 0–1 matrix is n 2  n þ 1: Interpreting non- singularity as full rank, we may ask further the question: What are the possible numbers of ones in a 0–1 matrix with given rank? We will answer this question in the generic case and in the symmetric case. The rank 2 symmetric case is subtle. Valiant [2] defined the rigidity R A (k) of a matrix A to be the minimal number of entries in the matrix that have to be changed in order to reduce the rank of A to less than or equal to k. So our work is along lower bounds on rigidity of explicit matrices. See [3]. In section 2 we prove the main results. In section 3 we give some examples. *Corresponding author. Email: zhan@math.ecnu.edu.cn Linear and Multilinear Algebra ISSN 0308-1087 print: ISSN 1563-5139 online ß 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/03081080500094048