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 $x_1 + . . . + x_n$ be? The answers to these two celebrated questions, asked by Erdős and Lovász, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lovász $\vartheta$-function and minimum semidefinite rank. In this talk we will study these parameters for general $H$-free graphs. In particular, we show that for certain bipartite graphs $H$, there is a connection between the Turán number of $H$ and the maximum of $\vartheta(\overline{G})$ over all $H$-free graphs.
JJoint work with Shoham Letzter and Benny Sudakov.