Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
18
The categories of this paper are: econ
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas