A note on traces of set families (Q1937361)

Let \(\mathcal F\) be a family of subsets of \(\{1,\ldots,n\}\). Let \(k>l>0\) and \(l<n\). The author shows that if for any \((n-l)\)-subset \(L\) of \(\{1,\ldots,n\}\), the family of traces \(\mathcal F|_{L}=\{F\cap L:\;F\in\mathcal F\}\) does not contain any chain of length \(k+1\), then \[ |\mathcal F|\leq (k-l+o(1))\binom n{\lfloor n/2\rfloor}, \] and the upper bound is asymptotically tight.