Weak regularity and finitely forcible graph limits

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e., a graphon determined by finitely many subgraph densities, is simple structured. 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−2/25log⁎⁡ε−2. This bound almost matches the known upper bound and, in a certain sense, is the best possible.