1-intersecting families (Q1062056)

The authors study combinatorial properties of subset families yielded by graph (or hypergraph) substitution decomposition and also by others combinatorial decomposition theories such as those studied by Cunningham and Edmonds. The main family \({\mathcal F}\) studied here (called 1- intersecting family) is made with subsets of a ground set X and satisfies: \(\emptyset,X\in {\mathcal F}\), for all \(A\in {\mathcal F}| A| \geq 2\), \(A\in {\mathcal F}\Rightarrow \bar A\in {\mathcal F}\), \(| A\cap B| >1\) and \(A\cup B\neq X\Rightarrow A\cap B\in {\mathcal F}.\) A characterization theorem and a recursive construction of 1-intersecting family are given.