Almost quadratic gap between partition complexity and query/communication complexity

Abstract: We show nearly quadratic separations between two pairs of complexity measures: 1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D<sup>{sc}(f))<sup>{2-o(1)})$ where $D(f)$ is the deterministic query complexity of $f$ and $D<sup>{sc}$ is the subcube partition complexity of $f$; 2. As a consequence, we obtain that there is a communication task $f(x, y)$ such that $D<sup>{cc}(f)=\Omega(\log<sup>{2-o(1)}\chi(f))$ where $D<sup>{cc}(f)$ is the deterministic 2-party communication complexity of $f$ (in the standard 2-party model of communication) and $\chi(f)$ is the partition number of $f$. Both of those separations are nearly optimal: it is well known that $D(f)=O((D<sup>{sc}(f))<sup>{2})$ and $D<sup>{cc}(f)=O(\log<sup>2\chi(f))$.