Let x1,…,xn∈Rd 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 x1+…+xn 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 ϑ-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 ϑ(G) over all H-free graphs G.Comment: 16 page