VC-dimensions for set familes between partially ordered set and totally ordered set (2412.06402v2)

Abstract: We say that two partial orders on $[n]$ are compatible if there exists a partial order that is finer than both of them. Under this compatibility relation, the set of all partial orders $\mathcal{F}$ and the set of all total orders $\mathcal{G}$ on $[n]$ naturally define set families on each other, where each order is identified with the set of orders that are compatible with it. In this note, we determine the VC-dimension of $\mathcal{F}$ on $\mathcal{G}$ by showing that $\operatorname{VC}{\mathcal{G}}(\mathcal{F}) = \lfloor\frac{n^2}{4}\rfloor$ for $n \ge 4$. We also prove $2(n-3) \le \operatorname{VC}{\mathcal{F}}(\mathcal{G}) \le n \log_2 n$ for $n \ge 1$.