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Uniform rectangle lower bounds from randomized complexity

Determine whether there exists a function κ: ℕ → (0,1) such that for every Boolean matrix F, the rectangle ratio satisfies rect(F) ≥ κ(R(F)), where R(F) is the public-coin two-sided error randomized communication complexity.

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Background

Understanding whether bounded randomized complexity enforces large monochromatic rectangles would provide structural insights into BPP and potentially relate it to oracle-aided deterministic classes via known lower-bound methods.

Current techniques do not yield any nontrivial uniform dependence, and even superweak bounds are unknown.

References

In fact, we do not know whether there is a uniform lower bound on $rect(F)$ depending only on $R(F)$.

— Structure in Communication Complexity and Constant-Cost Complexity Classes (2401.14623 - Hatami et al., 26 Jan 2024) in After Question on rectangles vs BPP, Section 5.1 (The power of randomness: BPP)