The ExactlyN problem in the number-on-forehead (NOF) communication setting asks $k$ players, each of whom can see every input but their own, if the $k$ input numbers add up to $N$. Introduced by Chandra, Furst and Lipton in 1983, ExactlyN is important for its role in understanding the strength of randomness in communication complexity with many players. It is also tightly connected to the field of combinatorics: its $k$-party NOF communication complexity is related to the size of the largest corner-free subset in $[N]^{k-1}$. In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for 3 players. As an immediate consequence, this also gave a new construction of larger corner-free subsets in $[N]^2$. Later that year Green gave a further refinement to their argument. These results represent the first improvements to the highest-order term for $k=3$ since the famous work of Behrend in 1946. In this paper we give a corresponding improvement to the highest-order term for all $k>3$, the first since Rankin in 1961. That is, we give a more efficient protocol for ExactlyN as well as larger corner-free sets in higher dimensions. Nearly all previous results in this line of research approached the problem from the combinatorics perspective, implicitly resulting in non-constructive protocols for ExactlyN. Approaching the problem from the communication complexity point of view and constructing explicit protocols for ExactlyN was key to the improvements in the $k=3$ setting. As a further contribution we provide explicit protocols for ExactlyN for any number of players which serves as a base for our improvement.

翻译：在额头数（NOF）通信设置中，ExactlyN问题要求$k$名玩家（每位玩家可看到除自身外的所有输入）判断$k$个输入数字之和是否为$N$。该问题由Chandra、Furst和Lipton于1983年提出，因其在理解多玩家通信复杂度中随机性的作用而具有重要意义。同时它与组合学紧密相关：其$k$方NOF通信复杂度与$[N]^{k-1}$中最大无角子集的大小相关。2021年，Linial与Shraibman给出了针对3玩家ExactlyN问题的高效协议，并由此直接构造出$[N]^2$中更大的无角子集。同年晚些时候，Green进一步改进了他们的论证。这些结果标志着自1946年Behrend开创性工作以来，$k=3$情况下最高阶项的首个改进。本文针对所有$k>3$的情况给出了相应的高阶项改进（自1961年Rankin工作以来首次），即我们不仅给出了更高效的ExactlyN协议，还构造了更高维度中的更大无角子集。此前该研究方向几乎所有成果均从组合学角度处理问题，隐式地导致ExactlyN协议的非构造性。而从通信复杂度视角出发并显式构造ExactlyN协议，正是$k=3$情形取得改进的关键。作为进一步贡献，我们为任意数量玩家提供了显式ExactlyN协议，这构成了本改进的基础。