Randomized Communication and Implicit Representations for Matrices and Graphs of Small Sign-Rank

We prove a characterization of the structural conditions on matrices of sign-rank 3 and unit disk graphs (UDGs) which permit constant-cost public-coin randomized communication protocols. Therefore, under these conditions, these graphs also admit implicit representations. The sign-rank of a matrix $M \in \{\pm 1\}^{N \times N}$ is the smallest rank of a matrix $R$ such that $M_{i,j} = \mathrm{sign}(R_{i,j})$ for all $i,j \in [N]$; equivalently, it is the smallest dimension $d$ in which $M$ can be represented as a point-halfspace incidence matrix with halfspaces through the origin, and it is essentially equivalent to the unbounded-error communication complexity. Matrices of sign-rank 3 can achieve the maximum possible bounded-error randomized communication complexity $\Theta(\log N)$, and meanwhile the existence of implicit representations for graphs of bounded sign-rank (including UDGs, which have sign-rank 4) has been open since at least 2003. We prove that matrices of sign-rank 3, and UDGs, have constant randomized communication complexity if and only if they do not encode arbitrarily large instances of the Greater-Than communication problem, or, equivalently, if they do not contain arbitrarily large half-graphs as semi-induced subgraphs. This also establishes the existence of implicit representations for these graphs under the same conditions.