Singularity of $\{\pm 1\}$-matrices and asymptotics of the number of threshold functions

Abstract

Two results concerning the number of threshold functions P (2, n) and the probability P n that a random n×n Bernoulli matrix is singular are established. We introduce a supermodular function η ⋆ n : 2 RP n f in → Z ≥0 , defined on finite subsets of RP n , that allows us to obtain a lower bound for P (2, n) in terms of P n+1 . This, together with L. Schläfli's famous upper bound, give us asymptoticsAlso, the validity of the long-standing conjecture concerning P n is proved: