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Large monochromatic rectangles in BPP

Establish whether every N×N communication matrix whose problem lies in BPP contains monochromatic rectangles of size Ω(N)×Ω(N).

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Background

Large monochromatic rectangles are a hallmark of certain efficient communication protocols. The authors highlight that all known BPP matrices possess such large rectangles, and prior works conjecture this is a universal phenomenon. Proving or refuting this would impact understanding of structural properties of BPP and connections to one-sided error protocols.

References

Also, all known N × N matrices in BPP have monochromatic rectangles of size Ω(N) × Ω(N), and [CLV19, HHH22dimfree] conjecture that this is always true (it is true for problems solved by (1) queries to RP0). We mentioned above that finding a complete problem (or hierarchy) could have answered many questions about BPP (depending on the properties of the complete problem itself); these two open problems are examples of that.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Section: Constant-Cost Communication: the Story so Far and Farther