## Abstract

For all integers n≥k>d≥1, let m

γ > 0, we show that the same conclusion holds if H is an n-vertex k-uniform hypergraph with δ

_{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+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 andγ > 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.Original language | English |
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Number of pages | 34 |

Journal | Combinatorica |

Early online date | 5 Aug 2024 |

DOIs | |

Publication status | E-pub ahead of print - 5 Aug 2024 |