Infinite‐dimensional finitely forcible graphon

Abstract

Graphons are analytic objects associated with convergent sequences of dense graphs. Finitely forcible graphons, that is, those determined by finitely many subgraph densities, are of particular interest because of their relation to various problems in extremal combinatorics and theoretical computer science. Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon always has finite dimension, which would have implications on the minimum number of parts in its weak ε‐regular partition. We disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices that has infinite dimension.

On the boundary of the region defined by homomorphism densities

“…As mentioned in the introduction, such problems can be extremely difficult in full generality. Increasingly complex finitely forcible graphons are constructed in [GGKK15, GKK14,LS11], furthering the evidence in support of this claim. In particular, it is asked in [LS11] for which graphs F the infinite lexicographic power ⊗ ∞ F is finitely forcible.…”

Compactness and finite forcibility of graphons

“…However, the dimension is finite when several other notions of dimension are considered, and so we do not claim to disprove this conjecture in this paper. In [22], the first two authors and Klimošová disprove Conjecture 2 in a more convincing way: they construct a finitely forcible graphon W such that a subspace of T (W ) is homeomorphic to [0, 1] ∞ . The graphon constructed in [22] also has infinite Minkowski dimension with respect to the metric d W , which has implications on the sizes of its weak regularity partitions [35,40].…”

“…This framework can be used to construct finitely forcible graphons with other non-trivial properties. In particular, it was used in [22] to completely disprove Conjecture 2 (see Section 7 for further details), and in [14] to construct finitely forcible graphons with no small weak regularity partitions. This line of research culminated with [13] and [31], where the techniques set out in this paper were used to show that any graphon can be a subgraphon of a finitely forcible graphon.…”

Finitely forcible graph limits are universal

“…The set of all vertices with degree d i will be referred to as a part; the size of a part is its measure and its degree is the common degree of its vertices. The following lemma was proved in [15,16]. Lemma 4.…”

“…Early examples of finitely forcible graphons indicated that all finitely forcible graphons might possess a simple structure, as formalized by Lovász and Szegedy, who conjectured the following [24, Conjectures 9 and 10]. Both conjectures were disproved through counterexample constructions [15,16]. A stronger counterexample to Conjecture 2 was found in [12]: Conjecture 2 would imply that the number of parts of a weak ε-regular partition of a finitely forcible graphon is bounded by a polynomial of ε −1 but the construction given in [12] almost matches the best possible exponential lower bound from [11].…”