Exact results on traces of sets

For non-negative integers $n$, $m$, $a$ and $b$, we write $(n,m)\to (a,b)$ if for every family $\mathcal{F}\subseteq 2^{[n]}$ with $\mathrm{\mid }\mathcal{F}\mathrm{\mid }⩾m$ there is an $a$-element set $T\subseteq [n]$ such that $\mid {\mathcal{F}}_{\mid T}\mid ⩾b$, where ${\mathcal{F}}_{\mid T}=\{F\cap T:F\in \mathcal{F}\}$. A longstanding problem in extremal set theory asks to determine $m(s)={\lim }_{n\to +\mathrm{\infty }}\frac{m(n,s)}{n}$, where $m(n,s)$ denotes the maximum integer $m$ such that $(n,m)\to (n-1,m-s)$ 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 $$m(n,2^{d-1}-c)=\frac{2^d-c}{d}nfor1\le c\le d-1andd\mid n,andm(n,2^{d-1}-d)=\frac{2^d-d-0.5}{d}nfor2d\mid n$$ holds for $d\ge 50$. Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that $m(11)=5.3$.