Largest $3$-uniform set systems with VC-dimension $2$

TL;DR

The paper solves the extremal problem for $3$-uniform set systems with VC-dimension at most $2$ on ground sets $[n]$, showing the maximum size equals $\binom{n-1}{2}+1$ for all $n\ge 7$, with $\binom{n}{3}$ for $n\le 5$, and $13$ when $n=6$. The authors develop a witness-based decomposition, introducing the auxiliary structures $\mathcal{B}$ (2-element witnesses), $L$ (singleton witnesses), and $\mathcal{C}$ (empty witnesses), and prove a key identity $|\mathcal{F}|=|\mathcal{B}|+|L|+|\mathcal{C}|$. The proof proceeds by induction on $n$, with a base case established computationally for $n=7$, and a two-pronged analysis depending on whether $\mathcal{C}$ is empty or not; in both branches, a detailed case analysis and forbiddance of certain substructures (e.g., 2-intersecting families, linear triangles) lead to the tight bound. These results settle the $d=2$ instance of a central extremal question, connect to Mubayi–Zhao constructions, and motivate continued exploration of extremal configurations and improved bounds for larger $d$.

Abstract

We determine the largest size of $3$-uniform set systems on $[n]$ with VC-dimension $2$ for all $n$.

Theorems & Definitions (36)

• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• Lemma 2.3: 2025CombProof
• Claim 2.4
• proof : Proof of claim
• Definition 2.5
• Claim 2.6
• proof : Proof of claim
• Claim 2.7