Perfect matchings in random sparsifications of Dirac hypergraphs

For all integers n≥k>d≥1, let m_d(k,n) be the minimum integer D≥0 such that every k-uniform n-vertex hypergraph H with minimum d-degree δ_d(H) at least D has an optimal matching. For every fixed integer k≥3, we show that for n∈kN and p=Ω(n^−k+1log n), if H is an n-vertex k-uniform hypergraph with δ_k−1(H)≥m_k−1(k,n), then a.a.s.\ its p-random subhypergraph H_p contains a perfect matching. Moreover, for every fixed integer d < k and 
γ > 0, we show that the same conclusion holds if H is an n-vertex k-uniform hypergraph with δ_d(H)≥m_d(k,n)+γ(^n−d_k−d). Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as ``robust'' versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, H has at least exp((1−1/k)nlogn−Θ(n)) many perfect matchings, which is best possible up to an exp(Θ(n)) factor.