Orthonormal Representations of H-Free Graphs

Igor Balla, Shoham Letzter, Benny Sudakov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 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.

Original languageEnglish
Pages (from-to)654-670
Number of pages17
JournalDiscrete and Computational Geometry
Issue number3
StatePublished - 1 Oct 2020
Externally publishedYes


  • Lovász ϑ-function
  • Minrank
  • Orthonormal representation
  • Turán numbers


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