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Journal of the European Mathematical Society

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Volume 21, Issue 10, 2019, pp. 3199–3223
DOI: 10.4171/JEMS/901

Published online: 2019-06-24

Compactness and finite forcibility of graphons

Roman Glebov[1], Daniel Král'[2] and Jan Volec[3]

(1) Ben Gurion University of the Negev, Beer-Sheva, Israel
(2) Masaryk University, Brno, Czech Republic, and University of Warwick, Coventry, UK
(3) Emory University, Atlanta, USA

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.

Keywords: Graph limits, extremal combinatorics

Glebov Roman, Král' Daniel, Volec Jan: Compactness and finite forcibility of graphons. J. Eur. Math. Soc. 21 (2019), 3199-3223. doi: 10.4171/JEMS/901