Large monochromatic rectangles in constant-cost randomized communication (BPP)

Establish whether every communication problem that admits a constant-cost public-coin randomized protocol (the class BPP as defined in this paper) has linear-size monochromatic rectangles; specifically, prove or refute that for each matrix P_N in such a problem, there exists a monochromatic submatrix of size Ω(N) × Ω(N).

Background

A central technique for proving separations in communication complexity relies on the existence of large monochromatic rectangles. For the class of constant-cost public-coin randomized protocols (BPP in this paper's sense), the structure of matrices is not well understood, and lower-bound methods that use rectangles are hindered by the lack of general guarantees. The authors emphasize that resolving the existence of linear-size rectangles would be a foundational step toward many results, including oracle lower bounds.