Contradiction Graphs Determine VC Dimension

arXiv:2605.20434v1 Announce Type: new
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).