On uniform intersection numbers (Q2715956)

An intersection graph \(G(F)\) of a family \(F\) of sets is a graph whose vertex set is \(F\) and in which two vertices are adjacent if and only if their intersection (as sets) is non-empty. The intersection number of a graph \(G\) is the minimum number of sets of a family \(F\) such that \(G\cong G(F)\). Similarly the uniform intersection number of \(G\) is defined; in this case an additional condition is required that all sets of \(F\) have equal cardinality. The paper studies these numbers and also their variants in whose definitions multifamilies instead families are considered; in a multifamily the same set may occur more than once.