Rainbow Matchings in r-Partite r-Graphs

Given a collection of matchings M = (M1,M2, . . . ,Mq) (repetitions allowed), a matching M contained in ⋃ M is said to be s-rainbow for M if it contains representatives from s matchings Mi (where each edge is allowed to represent just one Mi). Formally, this means that there is a function φ : M → [q] such that e ∈ Mφ(e) for all e ∈ M , and |Im(φ)| > s. Let f(r, s, t) be the maximal k for which there exists a set of k matchings of size t in some r-partite hypergraph, such that there is no s-rainbow matching of size t. We prove that f(r, s, t) > 2r−1(s − 1), make the conjecture that equality holds for all values of r, s and t and prove the conjecture when r = 2 or s = t = 2. In the case r = 3, a stronger conjecture is that in a 3-partite 3-graph if all vertex degrees in one side (say V1) are strictly larger than all vertex degrees in the other two sides, then there exists a matching of V1. This conjecture is at the same time also a strengthening of a famous conjecture, described below, of Ryser, Brualdi and Stein. We prove a weaker version, in which the degrees in V1 are at least twice as large as the degrees in the other sides. We also formulate a related conjecture on edge colorings of 3-partite 3-graphs and prove a similarly weakened version.