Linear-size monochromatic rectangles in constant-cost randomized communication (BPP)

Determine whether every communication problem in the constant-cost public-coin class BPP necessarily has linear-size monochromatic rectangles. Concretely, ascertain whether there exists a universal constant alpha > 0 such that for every matrix P_N in any problem P ∈ BPP, there are row and column subsets R, C ⊆ [N] with |R|, |C| ≥ alpha·N for which the submatrix P_N[R×C] is monochromatic.

Background

Monochromatic rectangle structure is a central tool for proving lower bounds in communication complexity. For constant-cost randomized communication (denoted BPP in the paper), many known lower-bound techniques rely on large monochromatic rectangles, but the structural theory of BPP is still limited.

The authors emphasize that even the basic question of whether all BPP problems admit large (linear-size) monochromatic rectangles is unresolved, which hinders rectangle-based techniques (e.g., η-area) from being applied to prove separations inside BPP.