VC-dimensions for set familes between partially ordered set and totally ordered set

TL;DR

This paper studies the VC-dimension of the set of all partial orders $\mathcal{F}$ on $[n]$ as a subset of the total orders $\mathcal{G}$, and vice versa, under the compatibility condition that $<_1 \cup <_2$ is acyclic. It derives an exact bound $\operatorname{VC}_{\mathcal{G}}(\mathcal{F}) = \left\lfloor \frac{n^2}{4} \right\rfloor$ for $n \ge 4$ and establishes $2(n-3) \le \operatorname{VC}_{\mathcal{F}}(\mathcal{G}) \le n \log_2 n$ for $n \ge 1$. The proofs rely on representing orders as directed graphs, using acyclicity arguments to bound shattered sets, and constructing explicit shattered families on $\mathcal{G}$ and combinatorial lower-bound constructions via $(G_i)$ and $(H_i)$. These results illuminate the VC-dimension behavior of interplaying order families and connect to the Kleitman-Rothschild asymptotics for the number of partial orders.

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 $n \ge 1$.

Figures (1)

• Figure :

Theorems & Definitions (6)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of \ref{['thm:vc_dim_partial_to_total']}
• proof : Proof of \ref{['thm:vc_dim_total_to_partial']}