针对多于3名玩家的ExactlyN问题的一种改进协议
An improved protocol for ExactlyN with more than 3 players
AI总结：
本文针对多于3名玩家的ExactlyN问题，从通信复杂度视角构造显式协议，首次自1961年以来改进了最高阶项，并由此得到更高维度的更大无角集。
摘要：
在额上数字（NOF）通信模型中，ExactlyN问题询问k名玩家（每人都能看到除自己外的所有输入），这k个输入数字之和是否等于N。该问题由Chandra、Furst和Lipton于1983年提出，因其在理解多方通信复杂度中随机性作用方面的重要性而备受关注。它与组合数学紧密相关：其k方NOF通信复杂度与[N]^{k-1}中最大无角子集的大小有关。2021年，Linial和Shraibman为3名玩家提供了更高效的ExactlyN协议，直接在[N]^2中构造出更大的无角子集。同年Green对其论证作了进一步改进。这是自Behrend 1946年的著名工作以来，对k=3时最高阶项的首次改进。本文针对所有k>3的情况，对最高阶项给出了相应改进，这是自Rankin 1961年以来的首次。即，我们为ExactlyN提供了更高效的协议，并在更高维度获得了更大的无角集。此前该领域几乎所有研究均从组合学视角出发，隐式地得到非构造性协议。从通信复杂度视角出发并为ExactlyN构造显式协议，是k=3取得改进的关键。作为进一步贡献，我们为任意玩家数提供了显式协议，以此作为我们改进的基础。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
