Weak regularity and finitely forcible graph limits

TL;DR

This paper constructs a finitely forcible graphon with a weak regularity partition requiring exponentially many parts, nearly matching the known bounds for graphs and challenging existing conjectures about graphon structure.

Contribution

It provides a counterexample to the conjecture that finitely forcible graphons have simple regular partitions with polynomially bounded parts.

Findings

Constructed a finitely forcible graphon with exponential lower bound on regular partition size.

Showed the lower bound nearly matches the upper bound for graphs.

Challenged the conjecture about simple structure of finitely forcible graphons.

Abstract

Graphons are analytic objects representing limits of convergent sequences of graphs. Lov\'asz 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 $\varepsilon$-regular partition with the number of parts bounded by a polynomial in $\varepsilon^{-1}$. We construct a finitely forcible graphon $W$ such that the number of parts in any weak $\varepsilon$-regular partition of $W$ is at least exponential in $\varepsilon^{-2}/2^{5\log^*\varepsilon^{-2}}$. This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.