Weak regularity and finitely forcible graph limits

Abstract

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many graph densities, has a simple structure. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak ε-regular partition with the number of parts bounded by a polynomial in ε −1 . We construct a finitely forcible graphon W such that the number of parts in any weak ε-regular partition of W is at least exponential in ε −2 /2 5 log * ε −2 . This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.

Infinite‐dimensional finitely forcible graphon

“…First, Cooper et al . addressed one of the motivations for Conjecture and constructed a finitely forcible graphon such that the number of parts in every weak ‐regular partition of is at least for an infinite sequence of tending to 0. This almost matches the general upper bound of on the number of parts in weak ‐regular partitions .…”

“…The method for establishing that a graphon is finitely forcible using decorated constraints, which originated in [18] and was further developed in this paper, turned out to be useful in several follow up results, which we now mention. First, Cooper et al [12] addressed one of the motivations for Conjecture 1 and constructed a finitely forcible graphon W such that the number of parts in every weak ε-regular partition of W is at least 2 Ω(ε −2 /2 5 log * ε −2 ) for an infinite sequence of ε tending to 0. This almost matches the general upper bound of 2 Θ(log 2 ε −1 ) on the number of parts in weak ε-regular partitions [16].…”

“…The second constraint in Figure 19 implies deg B 1,k b = deg B 4,k b + deg B 5,k b for almost every b ∈ B 1,k . Hence, it must hold that N B 1,k (b) is the set given in (12) and W (b, b ) = 1 for almost every b ∈ B 1,k and b ∈ N B 1,k (b). We conclude that the graphons W and W ψ agree almost everywhere on B 1,k × B 1,k for every k ∈ N.…”

“…One of the contributions of this paper is showing how the techniques from [] and Norine [Private Communication] can be refined to force a subspace homeomorphic to , which turned out to be quite challenging. Another contribution of the paper is formalizing the methods used in [] and Norine [Private Communication], which are further used in the follow up papers .…”

Extremal graph theory and finite forcibility

“…It was believed that every finitely forcible graphon has a simple structure [12, Conjectures 9 and 10]. However, this is not the case in quite a strong sense [5], also see [4,6,7]. …”