Asymptotic enumeration of incidence matrices

We discuss the problem of counting {\em incidence matrices}, i.e. zero-one matrices with no zero rows or columns. Using different approaches we give three different proofs for the leading asymptotics for the number of matrices with $n$ ones as $n\to \mathrm{\infty }$. We also give refined results for the asymptotic number of $i\times j$ incidence matrices with $n$ ones.