Almost disjoint triangles in 3-space (Q1864116)

Two triangles are called almost disjoint if their intersection is either empty or consists of one common vertex. Let \(f(n)\) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of \(n\) points in 3-space. A first construction of \(m^2+{m\choose 2}\) points which constitute the vertex set of \(m{m\choose 2}\) almost disjoint triangles shows that \(f(n)= \Omega(n^{3/2})\). A second construction based on cyclic polytopes confirms that result and leads to an open question: Is there, for \(n\) arbitrarily large, a closed polyhedral manifold in 3-space with \(n\) vertices whose genus is \(\Omega(n^{3/2})\)?