The log-rank conjecture for deterministic communication complexity

Let $f:X\times Y\to\{0,1\}$ be a Boolean function, and let $\mathrm{CC}^{\mathrm{det}}(f)$ denote its deterministic communication complexity. Let $M_f$ be its communication matrix and write $\mathrm{rank}(f)=\mathrm{rank}(M_f)$ over the reals. The log-rank conjecture. There exists a universal constant $C>0$ such that, for every Boolean function $f$,

$$\mathrm{CC}^{\mathrm{det}}(f)\le C(\log\mathrm{rank}(f))^C.$$

The rank lower bound gives the converse inequality up to the choice of logarithm, and the conjecture asks whether this lower bound is tight up to polynomial factors. It was proposed by Lovász and Saks and remains a fundamental open problem in communication complexity.