On extremal problems concerning the traces of sets

TL;DR

This paper investigates extremal hypergraph problems related to vertex traces, providing new formulas and bounds for the maximum number of edges under certain trace conditions, extending previous results for specific parameters.

Contribution

It establishes new exact formulas for m(n,s) when s is close to powers of two, and solves cases for small s, including a conjecture by Frankl and Watanabe.

Findings

Proves m(n,2^{d-1}-c) = (n/d)(2^d - c) for d ≥ 4c and d | n.

Determines m(n,2^{d-1}-c) for c=3,4, solving more small s cases.

Provides a counterexample showing the formula does not hold for c=d.

Abstract

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.