Orthonormal Representations of H-Free Graphs

Let x 1 , … , x n ∈ 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 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 Turán number of H and the maximum of ϑ ( G ¯ ) over all H -free graphs G .