We prove that the list replicability number of $d$-dimensional $\gamma$-margin half-spaces satisfies \[ \frac{d}{2}+1 \le \mathrm{LR}(H^d_\gamma) \le d, \] which grows with dimension. This resolves several open problems: $\bullet$ Every disambiguation of infinite-dimensional large-margin half-spaces to a total concept class has unbounded Littlestone dimension, answering an open question of Alon, Hanneke, Holzman, and Moran (FOCS '21). $\bullet$ Every disambiguation of the Gap Hamming Distance problem in the large gap regime has unbounded public-coin randomized communication complexity. This answers an open question of Fang, G\"o\"os, Harms, and Hatami (STOC '25). $\bullet$ There is a separation of $O(1)$ vs $\omega(1)$ between randomized and pseudo-deterministic communication complexity. $\bullet$ The maximum list-replicability number of any finite set of points and homogeneous half-spaces in $d$-dimensional Euclidean space is $d$, resolving a problem of Chase, Moran, and Yehudayoff (FOCS '23). $\bullet$ There exists a partial concept class with Littlestone dimension $1$ such that all its disambiguations have infinite Littlestone dimension. This resolves a problem of Cheung, H. Hatami, P. Hatami, and Hosseini (ICALP '23). Our lower bound follows from a topological argument based on a local Borsuk-Ulam theorem. For the upper bound, we construct a list-replicable learning rule using the generalization properties of SVMs.

翻译：我们证明了 $d$ 维 $\gamma$ 间隔半空间的列表可复制数满足 \[ \frac{d}{2}+1 \le \mathrm{LR}(H^d_\gamma) \le d, \] 该数随维度增长。这解决了几个开放问题：$\bullet$ 无限维大间隔半空间到全概念类的任何消歧都具有无界的 Littlestone 维数，这回答了 Alon、Hanneke、Holzman 和 Moran（FOCS '21）提出的一个开放问题。$\bullet$ 在大间隔机制下，Gap Hamming Distance 问题的任何消歧都具有无界的公币随机通信复杂度。这回答了 Fang、Göös、Harms 和 Hatami（STOC '25）提出的一个开放问题。$\bullet$ 随机通信复杂度与伪确定性通信复杂度之间存在 $O(1)$ 与 $\omega(1)$ 的分离。$\bullet$ $d$ 维欧几里得空间中任意有限点集与齐次半空间的最大列表可复制数为 $d$，这解决了 Chase、Moran 和 Yehudayoff（FOCS '23）提出的一个问题。$\bullet$ 存在一个 Littlestone 维数为 $1$ 的部分概念类，其所有消歧都具有无限的 Littlestone 维数。这解决了 Cheung、H. Hatami、P. Hatami 和 Hosseini（ICALP '23）提出的一个问题。我们的下界基于局部 Borsuk-Ulam 定理的拓扑论证。对于上界，我们利用 SVM 的泛化性质构造了一个列表可复制学习规则。