Maximal intersecting families (Q1898053)

On an \(n\)-element underlying set every maximal intersecting set system \(\mathcal F\) has \(2^{n- 1}\) edges. If \(A\in {\mathcal F}\) is minimal in \(\mathcal F\) then the interchange of \(A\) with \(X\backslash A\) supplies another (maximal) intersecting family. Based upon this obvious fact the paper proves some easy results, like fixing two maximal intersecting families one can transform the first one into the second by a series of these interchangings (called switches). An (intersecting) family is regular if every point of the underlying set is covered by the same number of edges. This paper's main contribution is a construction of regular maximal intersecting families on underlying sets of cardinality \(2^a\) where \(a> 3\).