Orthonormal representations of $H$-free graphs

Let $x_1,\dots ,x_n\in {\mathbb{R}}^d$ 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 $x_1+\dots +x_n$ 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 $\vartheta \left( \overline{G} \right)$overall$H$-freegraphs$G$.