Lovász–Saks log-rank conjecture for Boolean functions

Let $f: \mathcal X \times \mathcal Y \to \{0,1\}$ be a Boolean function, let $M_f$ be its communication matrix, let $\operatorname{rank}(f)$ be the rank of $M_f$ over $\mathbb R$, and let $\operatorname{CC}_{\det}(f)$ be its deterministic communication complexity.

Log-rank conjecture. For every function $f: \mathcal X \times \mathcal Y \to \{0,1\}$,

$$\operatorname{CC}_{\det}(f) \leq \operatorname{polylog}(\operatorname{rank}(f)).$$

The conjecture asks whether every low-rank Boolean matrix can be partitioned into a small number of monochromatic rectangles. It is a central open problem in communication complexity; the best bound stated in the source is Lovett's $O(\sqrt{\operatorname{rank}(f)}\log(\operatorname{rank}(f)))$ upper bound.