A Rainbow r-Partite Version of the Erdős–Ko–Rado Theorem. (23rd January 2017) - AHARONI, RON; HOWARD, DAVID
Abstract : Let [ n ] r be the complete r -partite hypergraph with vertex classes of size n . It is an easy exercise to show that every set of more than ( k −1) n r −1 edges in [ n ] r contains a matching of size k . We conjecture the following rainbow version of this observation: if F 1, F 2, . . ., F k ⊆ [ n ] r are of size larger than ( k −1) n r −1 then there exists a rainbow matching, that is, a choice of disjoint edges f i ∈ F i . We prove this conjecture for r =2 and r =3.