  Two results concerning the number of threshold functions $P(2, n)$ and the
probability ${\mathbb P}_n$ that a random $n\times n$ Bernoulli matrix is
singular are established. We introduce a supermodular function
$\eta^{\bigstar}_n : 2^{{\bf RP}^n}_{fin} \to \mathbb{Z}_{\geq 0},$ defined on
finite subsets of ${\bf RP}^n,$ that allows us to obtain a lower bound for
$P(2, n)$ in terms of ${\mathbb P}_{n+1}.$ This, together with L.Schl\"afli's
famous upper bound, give us asymptotics $$P(2, n) \thicksim 2 {2^n-1 \choose
n},\quad n\to \infty.$$ Also, the validity of the long-standing conjecture
concerning ${\mathbb P}_n$ is proved: $$\mathbb{P}_n \thicksim (n-1)^22^{1-n},
\quad n\to \infty .$$
