  Let $x_1, \ldots, x_n \in \mathbb{R}^d$ be unit vectors such that among any
three there is an orthogonal pair. How large can $n$ be as a function of $d$,
and how large can the length of $x_1 + \ldots + x_n$ be? The answers to these
two celebrated questions, asked by Erd\H{o}s and Lov\'{a}sz, are closely
related to orthonormal representations of triangle-free graphs, in particular
to their Lov\'{a}sz $\vartheta$-function and minimum semidefinite rank. In this
paper, we study these parameters for general $H$-free graphs. In particular, we
show that for certain bipartite graphs $H$, there is a connection between the
Tur\'{a}n number of $H$ and the maximum of $\vartheta \left( \overline{G}
\right)$ over all $H$-free graphs $G$.
