  The minrank of a graph $G$ on the set of vertices $[n]$ over a field
$\mathbb{F}$ is the minimum possible rank of a matrix $M\in\mathbb{F}^{n\times
n}$ with nonzero diagonal entries such that $M_{i,j}=0$ whenever $i$ and $j$
are distinct nonadjacent vertices of $G$. This notion, over the real field,
arises in the study of the Lov\'asz theta function of a graph. We obtain tight
bounds for the typical minrank of the binomial random graph $G(n,p)$ over any
finite or infinite field, showing that for every field $\mathbb{F}=\mathbb
F(n)$ and every $p=p(n)$ satisfying $n^{-1} \leq p \leq 1-n^{-0.99}$, the
minrank of $G=G(n,p)$ over $\mathbb{F}$ is $\Theta(\frac{n \log (1/p)}{\log
n})$ with high probability. The result for the real field settles a problem
raised by Knuth in 1994. The proof combines a recent argument of Golovnev,
Regev, and Weinstein, who proved the above result for finite fields of size at
most $n^{O(1)}$, with tools from linear algebra, including an estimate of
R\'onyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of
polynomials.
