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