Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Orthonormal representations of $H$-free graphs   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 $G$.

3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin