We study the contradiction graphs associated with binary concept classes. For a class $H \subseteq \{0,1\}^X$, the order-$m$ contradiction graph $G_m(H)$ has as vertices the $H$-realizable labeled sequences of length $m$, with two vertices adjacent when the two sequences assign opposite labels to some common domain point. Our main result is that the single graph $G_m(H)$ determines the threshold predicate $\mathrm{VCdim}(H)\ge m$. Consequently, the full sequence $(G_m(H))_{m \ge 1}$ determines the exact VC dimension and, in particular, detects finite versus infinite VC dimension, answering a question posed by Alon et al. (2024).

翻译：我们研究与二元概念类相关的矛盾图。对于类别 $H \subseteq \{0,1\}^X$，阶数为 $m$ 的矛盾图 $G_m(H)$ 的顶点为长度为 $m$ 的 $H$ 可实现标记序列，当两个序列对某个公共域点赋予相反标记时，这两个顶点相邻。我们的主要结果是：单个图 $G_m(H)$ 决定了阈值谓词 $\mathrm{VCdim}(H)\ge m$。因此，完整序列 $(G_m(H))_{m \ge 1}$ 决定了精确的VC维，并特别地检测有限与无限VC维，这回答了 Alon 等人（2024）提出的一个问题。