An ensemble of high rank matrices arising from tournaments

Suppose Fis a field and let a:=(a(1), a(2),...) be a sequence of non-zero elements in F. For (a)n:=( a(1),..., a(n)), we consider the family M-n(a) of n xnsymmetric matrices Mover Fwith all diagonal entries zero and the (i, j)th element of Meither a(i) or a(j) for i < j. In this short paper, we show that all matrices in a certain subclass of M-n(a)-which can be naturally associated with transitive tournaments-have rank at least left perpendicular2n/3right perpendicular - 1. We also show that if char(F) not equal 2and Mis a matrix chosen uniformly at random from M-n(a), then with high probability rank (M) >= (1/2- o(1))(n). (c) 2022 Elsevier Inc. All rights reserved.