Counterexamples to polylogarithmic-cost multiplicative density increments

Determine whether counterexamples exist to obtaining a multiplicative density increment for higher-complexity additive-combinatorial patterns while paying only a poly-logarithmic dependence on the density in the codimension of the associated substructures, and in particular whether such counterexamples distinguish higher-order cases from the multiplicative density-increment result established for the corners setting.

Background

The paper’s main corners bound is obtained through a multiplicative density-increment strategy, which improves the quantitative behavior of the density-increment process. The authors note that analogous strategies can be applied to patterns of higher complexity, but these methods currently require passing to comparatively small structured subobjects.

The unresolved issue is whether the costly dependence on density in the codimension of those substructures is intrinsic. A counterexample would separate the higher-order situation from the behavior achieved in the paper, whereas a positive result would support the broader applicability of efficient multiplicative density increments in additive combinatorics.