发表机构
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800; Univ. Lyon, ENS de Lyon, UCBL, CNRS, LIP（波尔多大学; 里昂大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造部分矩阵，其每个去歧义化都包含任意给定大小的二元矩阵，利用拉姆齐理论回答线性分类器去歧义化问题，并给出部分图对称结果。
AI 中文摘要
部分矩阵的条目属于集合 $\{0,1,\star\}$，去歧义化将每个 $\star$ 替换为 $0$ 或 $1$。我们构造了这样的部分矩阵：其完全指定的子矩阵满足强限制，然而每个去歧义化都包含指定大小的每个二元矩阵。我们的第一个结果回答了 Alon、Hanneke、Holzman 和 Moran 关于带间隔的线性分类器去歧义化的问题。对于 $0<\varepsilon<\pi/2$，令 $M_\varepsilon^d$ 为以单位球面 $\mathbb S^d$ 上的点为索引的部分矩阵，其中球面距离至多 $\varepsilon$ 的点对条目为 $0$，距离至少 $\pi-\varepsilon$ 的点对条目为 $1$，其余为 $\star$。尽管这些矩阵的 VC 维有界且不依赖于 $d$，我们证明一旦 $d$ 足够大，每个去歧义化都包含每个二元 $k\times k$ 矩阵。这也产生了一个 Littlestone 维为 $1$ 的部分概念类，其任何去歧义化都没有有限的 VC 维。我们还为每个 $k$ 构造了一个有限部分矩阵，其完全指定的 $2\times2$ 子矩阵全部是常数的，而每个去歧义化都包含每个二元 $k\times k$ 矩阵。一个对称的类似结果适用于部分图：对于每个 $k$，存在一个 VC 维至多 $1$ 的部分图，其完全指定的诱导子图全部是团或稳定集，然而每个去歧义化都包含每个 $k$ 顶点图作为诱导子图。去歧义化可以看作是对未指定条目的 $2$-着色，这使得拉姆齐理论成为强制规定模式的自然框架。我们的证明依赖于两个最近的拉姆齐定理：几何论证使用 Pálvölgyi 的稠密块定理，而组合构造依赖于 Reiher 和 Rödl 的围长拉姆齐定理，这是诱导拉姆齐定理的一个适当加强。
英文摘要
A partial matrix has entries in $\{0,1,\star\}$, and a disambiguation replaces each $\star$ by $0$ or $1$. We construct partial matrices whose fully specified submatrices satisfy strong restrictions, yet every disambiguation contains every binary matrix of a prescribed size. Our first result answers a question of Alon, Hanneke, Holzman and Moran on the disambiguation of linear classifiers with margin. For $0<\varepsilon<π/2$, let $M_\varepsilon^d$ be the partial matrix indexed by points of the unit sphere $\mathbb S^d$, with entry $0$ for pairs at spherical distance at most $\varepsilon$, $1$ for pairs at distance at least $π-\varepsilon$, and $\star$ otherwise. Although these matrices have VC-dimension bounded independently of $d$, we prove that every disambiguation contains every binary $k\times k$ matrix once $d$ is sufficiently large. This also yields a partial concept class of Littlestone dimension $1$ with no disambiguation of finite VC-dimension. We also construct, for every $k$, a finite partial matrix whose fully specified $2\times2$ submatrices are all constant, while every disambiguation contains every binary $k\times k$ matrix. A symmetric analogue holds for partial graphs: for every $k$, there exists a partial graph of VC-dimension at most $1$ whose fully specified induced subgraphs are all cliques or stable sets, yet every disambiguation contains every $k$-vertex graph as an induced subgraph. A disambiguation can be viewed as a $2$-coloring of the unspecified entries, making Ramsey theory a natural framework for forcing prescribed patterns. Our proofs draw on two recent Ramsey theorems: the geometric argument uses Pálvölgyi's Dense Block theorem, while the combinatorial constructions rely on the girth Ramsey theorem of Reiher and Rödl, a suitable strengthening of the induced Ramsey theorem.
