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 Schülke. To be precise, we show that \(m(n,2^{d-1}-c)={\left\{ \begin{array}{ll} \frac{2^{d}-c}{d}n & \text{ for } 1\leqslant c\leqslant d-1 \text{ and } d\mid n \\ \frac{2^{d}-d-0.5}{d}n & \text{ for } c=d \text{ and } 2d\mid n \end{array}\right. }\) holds for \(d\geqslant 50\) . Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that \(m(11)=5.3\) .
