  A fooling-set matrix has nonzero diagonal, but at least one in every pair of
diagonally opposite entries is 0. Dietzfelbinger et al. '96 proved that the
rank of such a matrix is at least $\sqrt n$. It is known that the bound is
tight (up to a multiplicative constant).
  We ask for the "typical" minimum rank of a fooling-set matrix: For a
fooling-set zero-nonzero pattern chosen at random, is the minimum rank of a
matrix with that zero-nonzero pattern over a field $\mathbb F$ closer to its
lower bound $\sqrt{n}$ or to its upper bound $n$? We study random patterns with
a given density $p$, and prove an $\Omega(n)$ bound for the cases when: (a) $p$
tends to $0$ quickly enough, (b) $p$ tends to $0$ slowly, and $|\mathbb
F|=O(1)$, (c) $p\in(0,1]$ is a constant.
  We have to leave open the case when $p\to 0$ slowly and $\mathbb F$ is a
large or infinite field (e.g., $\mathbb F=GF(2^n)$, $F=\mathbb{R}$).
