Rainbow perfect matchings in 3-partite 3-uniform hypergraphs

Let $m,n,r,s$ be nonnegative integers such that n≥ m=3r+s and 1≤ s≤ 3. Let \[δ(n,r,s)=\{{array}{ll} n^2-(n-r)^2 &if\ s=1 , \\[5pt] n^2-(n-r+1)(n-r-1) &if\ s=2,\\[5pt] n^2 - (n-r)(n-r-1) &if\ s=3. {array}.\] We show that there exists a constant n₀ > 0 such that if F₁,…, Fₙ are 3-partite 3-graphs with n≥ n₀ vertices in each partition class and minimum vertex degree of Fᵢ is at least δ(n,r,s)+1 for i ∈ [n] then ₁,…,Fₙ\ admits a rainbow perfect matching. This generalizes a result of Lo and Markstr\"om on the vertex degree threshold for the existence of perfect matchings in 3-partite 3-graphs. In this proof, we use a fractional rainbow matching theory obtained by Aharoni et al. to find edge-disjoint fractional perfect matching.