Log-rank conjecture for deterministic communication complexity

Prove that there exists a universal constant C > 0 such that for every Boolean matrix F, the deterministic communication complexity D(F) satisfies D(F) ≤ C (log(rank(F)))^C, where rank(F) denotes the real rank of F.

Background

Deterministic protocols partition a Boolean matrix into at most 2^{D(F)} monochromatic rectangles, yielding log(rank(F)) ≤ D(F) ≤ rank(F) + 1. The log-rank conjecture posits that the upper bound can be tightened to a polylogarithmic dependence on rank(F), giving a structural characterization of low-rank Boolean matrices via small deterministic communication complexity.

Despite decades of work, the best known upper bounds remain far from polynomial in log(rank(F)), and the conjecture remains a central open problem in communication complexity.