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.

翻译：我们刻画了符号秩为3的矩阵以及单位圆盘图（UDG）中允许常数代价公开硬币随机通信协议的结构条件。因此，在此条件下，这些图也具有内隐表示。矩阵$M \in \{\pm 1\}^{N \times N}$的符号秩是满足对所有$i,j \in [N]$有$M_{i,j} = \mathrm{sign}(R_{i,j})$的矩阵$R$的最小秩；等价地，它是$M$能表示为以通过原点的半空间为点-半空间关联矩阵的最小维数$d$，并且它本质上等价于无界错误通信复杂度。符号秩为3的矩阵可以实现最大可能的有界错误随机通信复杂度$\Theta(\log N)$，同时，关于有界符号秩图（包括符号秩为4的UDG）是否存在内隐表示的问题，至少自2003年以来一直悬而未决。我们证明：符号秩为3的矩阵和UDG具有常数随机通信复杂度，当且仅当它们不编码任意大规模的Greater-Than通信问题实例，或者等价地，当且仅当它们不包含任意大规模的半诱导子图形式的半图。这也建立了在这些条件下这些图存在内隐表示的结果。