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 ∈ k N and p = Ω ( n - k + 1 log 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 ) n log n - Θ ( n ) ) many perfect matchings, which is best possible up to an exp ( Θ ( n ) ) factor.