  The minrank over a field $\mathbb{F}$ of a graph $G$ on the vertex set
$\{1,2,\ldots,n\}$ is the minimum possible rank of a matrix $M \in
\mathbb{F}^{n \times n}$ such that $M_{i,i} \neq 0$ for every $i$, and
$M_{i,j}=0$ for every distinct non-adjacent vertices $i$ and $j$ in $G$. For an
integer $n$, a graph $H$, and a field $\mathbb{F}$, let $g(n,H,\mathbb{F})$
denote the maximum possible minrank over $\mathbb{F}$ of an $n$-vertex graph
whose complement contains no copy of $H$. In this paper we study this quantity
for various graphs $H$ and fields $\mathbb{F}$. For finite fields, we prove by
a probabilistic argument a general lower bound on $g(n,H,\mathbb{F})$, which
yields a nearly tight bound of $\Omega(\sqrt{n}/\log n)$ for the triangle
$H=K_3$. For the real field, we prove by an explicit construction that for
every non-bipartite graph $H$, $g(n,H,\mathbb{R}) \geq n^\delta$ for some
$\delta = \delta(H)>0$. As a by-product of this construction, we disprove a
conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by
questions in information theory, circuit complexity, and geometry.
