Asymptotic full-rainbow-matching conjecture for hypergraphs

Establish that, for every fixed integer r≥2, each of the four parameters g(r,n), g′(r,n), h(r,n), and h′(r,n) is asymptotic to n; equivalently, prove that the ratio of each parameter to n converges to 1 as n tends to infinity.

Background

The paper defines g(r,n) and g′(r,n) as the largest guaranteed rainbow matching size among n matchings of size n in r-partite and unrestricted r-uniform hypergraphs, respectively. It defines h(r,n) and h′(r,n) as the smallest matching size guaranteeing a rainbow matching of size n in the corresponding two settings.

The preceding results give nontrivial bounds for these parameters when r≥3. Conjecture 3.1 proposes that all four parameters have asymptotic density one relative to n for every fixed uniformity r.