The No-Clash Teaching Dimension is Bounded by VC Dimension

Non-clashing teaching minimizes teaching complexity among models satisfying the Goldman--Mathias collusion-avoidance criterion. While the Vapnik--Chervonenkis dimension (VCdim) measures the complexity of learning from samples, non-clashing teaching dimension (NCTD) measures the number of examples required when a teacher can select them. Whether NCTD is always bounded above by VCdim remains a fundamental open question in learning theory. In this paper, we investigate this question for the concept classes of closed neighborhoods in planar graphs. We prove a tight upper bound of two on their NCTD, thereby establishing the conjectured inequality for this family. For the positive variant NCTD+, in which teaching sets contain only positive examples, we obtain a tight upper bound of four. Finally, we show that deciding whether either NCTD+ or NCTD is at most one is NP-complete, even for concept classes of closed neighborhoods in planar graphs. In particular, despite the upper bound of two on NCTD for such classes, distinguishing between NCTD values of one and two remains NP-hard.

The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik--Chervonenkis (VC) dimension guarantees that every proper consistent learning rule is probably approximately correct (PAC). Blumer, Ehrenfeucht, Haussler, and Warmuth showed, assuming the Continuum Hypothesis, that the "well-behavedness" condition of the concept class cannot be omitted: they constructed a concept class of Borel sets of VC dimension one admitting a consistent learning rule that is not PAC. We show that the Continuum Hypothesis is unnecessary. Working in Zermelo--Fraenkel set theory with the Axiom of Choice (ZFC) alone, we construct a concept class of Borel sets on $[0,1]$ of VC dimension one and a proper consistent learning rule that is not PAC. More precisely, for a suitable Borel probability measure and target concept, the rule has true risk one at every sample size on a set of samples of outer probability one. Consequently, finite VC dimension and Borel measurability of the individual concepts do not suffice to guarantee that every proper consistent learning rule is PAC. The result shows, with no need of extra set-theoretical assumptions, that the additional regularity assumption in the fundamental theorem cannot in general be omitted.

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).