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Orthonormal Representations of H-Free Graphs

  • Ricky Pollack Memorial Issue
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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 \(x_1 + \cdots + x_n\) 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 \(\vartheta \)-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 Turán number of H and the maximum of \(\vartheta (\overline{G})\) over all H-free graphs G.

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Notes

  1. In [26], Lovász forgets to include the assumption that A is symmetric and \(A_{i,i} = 0\) for all i in his statement of Theorem 6, but it is clear that this is what he intended.

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Acknowledgements

We would like to thank Boris Bukh and Chris Cox for stimulating discussions.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Igor Balla & Benny Sudakov

  2. ETH Institute for Theoretical Studies, ETH Zurich, Zurich, Switzerland

    Shoham Letzter

  3. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Shoham Letzter

Authors
  1. Igor Balla
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  2. Shoham Letzter
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  3. Benny Sudakov
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Correspondence to Benny Sudakov.

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Dedicated to the memory of Ricky Pollack.

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S. Letzter: Research supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. B. Sudakov: Research supported in part by SNSF Grant 200021-175573.

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Balla, I., Letzter, S. & Sudakov, B. Orthonormal Representations of H-Free Graphs. Discrete Comput Geom 64, 654–670 (2020). https://doi.org/10.1007/s00454-020-00185-0

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