Forbidding complete hypergraphs as traces (Q2478165)

Let \(\mathcal H\) be a hypergraph on the finite underlying set \(X\) and \(X'\) be a subset of \(X\). The \textit{trace} of \(\mathcal H\) on \(X'\) is the hypergraph \(\{ H \cap X' : \forall H \in {\mathcal H}\}\) (each edge is considered just once). This paper studies the maximum number of edges in \(\mathcal H\) if it does not contain complete \(q\)-uniform hypergraph on \(t\) vertices as a trace. The authors solved the problem asymptotically (as \(| X| \rightarrow \infty\)) for several hypergraph classes.