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arXiv · 1905.01539

Orthonormal representations of $H$-free graphs

Igor Balla·Shoham Letzter·Benny Sudakov
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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 $G$.

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Igor Balla, Shoham Letzter, Benny Sudakov. 2019-05-04. Orthonormal representations of $H$-free graphs. https://arxiv.org/abs/1905.01539

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