Compactness and finite forcibility of graphons

  • Roman Glebov

    Ben Gurion University of the Negev, Beer-Sheva, Israel
  • Daniel Král'

    Masaryk University, Brno, Czech Republic, and University of Warwick, Coventry, UK
  • Jan Volec

    Emory University, Atlanta, USA
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Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. The construction method gives a general framework for constructing finitely forcible graphons with non-trivial properties.

Cite this article

Roman Glebov, Daniel Král', Jan Volec, Compactness and finite forcibility of graphons. J. Eur. Math. Soc. 21 (2019), no. 10, pp. 3199–3223

DOI 10.4171/JEMS/901