On extremal problems concerning the traces of sets

Given two non-negative integers $n$ and $s$, define $m(n,s)$ to be the maximal number such that in every hypergraph $\mathcal{H}$ on $n$ vertices and with at most $m(n,s)$ edges there is a vertex $x$ such that $|\mathcal{H}_x|\geq | E(\mathcal{H})| -s$, where $\mathcal{H}_x=\{H\setminus\{x\}:H\in E(\mathcal{H})\}$. This problem has been posed by F\"uredi and Pach and by Frankl and Tokushige. While the first results were only for specific small values of $s$, Frankl determined $m(n,2^{d-1}-1)$ for all $d\in\mathbb{N}$ with $d\mid n$. Subsequently, the goal became to determine $m(n,2^{d-1}-c)$ for larger $c$. Frankl and Watanabe determined $m(n,2^{d-1}-c)$ for $c\in\{0,2\}$. Other general results were not known so far. Our main result sheds light on what happens further away from powers of two: We prove that $m(n,2^{d-1}-c)=\frac{n}{d}(2^d-c)$ for $d\geq 4c$ and $d\mid n$ and give an example showing that this equality does not hold for $c=d$. The other line of research on this problem is to determine $m(n,s)$ for small values of $s$. In this line, our second result determines $m(n,2^{d-1}-c)$ for $c\in\{3,4\}$. This solves more instances of the problem for small $s$ and in particular solves a conjecture by Frankl and Watanabe.