Maximal intersection critical families of finite sets (Q1923504)

An \(r\)-clique is a pairwisely intersecting system of \(r\)-sets. This is maximal iff it is not contained by another \(r\)-clique with the same vertex set. An \(r\)-clique is intersection critical if no edge of it can be replaced with a smaller one preserving the intersection property. The paper proves that for any non-intersection critical, maximal \(r\)-clique \(\mathcal H\), which differs from \(K^r_{r+1}\), the property \(|{\mathcal H}|>|V({\mathcal H})|\) holds. It is also shown that the system of the lines of a finite projective plane, avoiding a fixed point, is a maximal intersection critical \(r\)-clique (but it is no maximal \(r\)-clique).