Contradiction Graphs Determine VC Dimension

Abstract

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).