Uniform Set Systems with Uniform Witnesses

Frankl--Pach and Erd\H{o}s conjectured that any $(d+1)$-uniform set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$ with VC-dimension at most $d$ has size at most $\binom{n-1}{d}$ when $n$ is sufficiently large. Ahlswede and Khachatrian showed that the conjecture is false by giving a counterexample of size $\binom{n-1}{d}+\binom{n-4}{d-2}$. For a set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$, the condition that its VC-dimension is at most $d$ can be reformulated as follows: for any $F\in\mathcal{F}$, there exists a set $B_F\subseteq F$ such that $F\cap F'\neq B_F$ for all $F'\in\mathcal{F}$. In this direction, the first author, Xu, Yip, and Zhang conjectured that the bound $\binom{n-1}{d}$ holds if we further assume that $|B_F|=s$ for every $F\in \mathcal{F}$ and for some fixed $0\leq s\leq d$. The case $s=0$ is exactly the Erd\H{o}s--Ko--Rado theorem, and the cases $s\in \{1,d\}$ were proved in the paper by the first author, Xu, Yip, and Zhang. In this short note, we show that the conjecture holds when $s\leq d/2$, and the maximal constructions are stars. Moreover, we construct non-star set families of size $\binom{n-1}{d}$ satisfying the condition for $d/2<s\leq d-1$, which suggests that the problem is substantially different in these cases.