Exact results on traces of sets

Abstract

For non-negative integers $n$, $m$, $a$ and $b$, we write $\left(n,m\right)\to \left(a,b\right)$ if for every family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|⩾m$ there is an $a$-element set $T\subseteq [n]$ such that $\left|{\mathcal{F}}_{\mid T}\right|⩾b$, where ${\mathcal{F}}_{\mid T}=\{F\cap T:F\in \mathcal{F}\}$. A longstanding problem in extremal set theory asks to determine $m(s)={\lim }_{n\to +\infty }\frac{m(n,s)}{n}$, where $m(n,s)$ denotes the maximum integer $m$ such that $\left(n,m\right)\to \left(n-1,m-s\right)$ holds for non-negatives $n$ and $s$. In this paper, we establish the exact value of $m(2^{d-1}-c)$ for all $1⩽c⩽d$ whenever $d⩾50$, thereby solving an open problem posed by Piga and Schülke. To be precise, we show that $$m(n,2^{d-1}-c)=\frac{2^{d}-c}{d}n \mbox{ for } 1\leq c\leq d-1 \mbox{ and } d\mid n, \mbox{ and } m(n,2^{d-1}-d)=\frac{2^{d}-d-0.5}{d}n \mbox{ for } 2d\mid n $$ holds for $d\ge 50$. Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that $m(11)=5.3$.