Efficient Randomized Communication Without Large Monochromatic Rectangles

Abstract: In this paper, we construct a total Boolean function with $\widetilde{O}(\log n)$ randomized communication protocol, while any monochromatic rectangle has density at most $O(2<sup>{-\mathrm{poly}(n)})$. As a corollary, it gives the first total function separation for $$\mathrm{BPP}\not\subseteq\mathrm{P}<sup>{\mathrm{NP}}$$ in the communication world. Inspired by Gavinsky's recent work (arXiv:2608.18784), our construction combines the cheat-sheet framework with fully linear PCPs.