# Orthonormal representations of $H$-free graphs

## TL;DR

This paper explores the limits of orthonormal representations of $H$-free graphs, connecting graph theory parameters like Turán numbers and Lovász $ heta$-function, and extends classical questions posed by Erdős and Lovász.

## Contribution

It generalizes the study of orthonormal representations and related parameters to $H$-free graphs, establishing new links between Turán numbers and Lovász $ heta$-function for bipartite graphs.

## Key findings

- Established bounds relating Turán numbers and Lovász $ heta$-function for $H$-free graphs.
- Extended classical results to broader classes of graphs beyond triangle-free.
- Provided new insights into the structure of $H$-free graphs and their orthonormal representations.

## Abstract

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 $G$.