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Explicit matrices with constant weighted rectangle ratio and unbounded sign-rank

Construct an explicit sequence of sign matrices F_n such that wrect(F_n)^{-1} = O(1) while the sign-rank satisfies lim_{n→∞} signrank(F_n) = ∞.

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Background

Existential results show that matrices can have bounded weighted rectangle ratio yet very large sign-rank, separating Rect_0 from UPP_0. However, no explicit constructions are known, and existing lower-bound techniques (VC-dimension, Forster’s method, rectangle bounds) fall short.

An explicit construction would expose the limitations of current sign-rank lower-bound machinery and clarify the landscape between Rect_0 and UPP_0.

References

The following problem is open. Construct an explicit sequence of matrices $F_n$ such that $wrect(F_n){-1}=O(1)$ and \lim_{n \to \infty} _{\pm}(F_n)=\infty.

— Structure in Communication Complexity and Constant-Cost Complexity Classes (2401.14623 - Hatami et al., 26 Jan 2024) in Problem 5.1, Section 5.3 (Sign-rank and UPP)