On perfect matchings and tilings in uniform hypergraphs^∗

In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for k ≥ 3, if H is a k-graph on n ∈ kN vertices with independence number at most n/p and minimum codegree at least (1/p + o(1))n, where p is the smallest prime factor of k, then H contains a perfect matching. Second, we show that if H is a 3-graph on n ∈ 3N vertices which does not contain any induced copy of K₄ ⁻ (the unique 3-graph with 4 vertices and 3 edges) and has minimum codegree at least (1/3 + o(1)))n, then H contains a perfect matching. Moreover, if we allow the matching to miss at most 3 vertices, then the minimum degree condition can be reduced to (1/6 + o(1)))n. Third, we show that if H is a 3-graph on n ∈ 4N vertices which does not contain any induced copy of K₄ ⁻ and has minimum codegree at least (1/8 + o(1)))n, then H contains a perfect Y -tiling, where Y represents the unique 3-graph with 4 vertices and 2 edges. We also provide the examples showing that our minimum codegree conditions are asymptotically best possible. Our main tool for finding the perfect matching is a characterization theorem in [J. Han, Trans. Amer. Math. Soc., 369 (2017), pp. 5197–5218] that characterizes the k-graphs with minimum codegree at least n/k which contain a perfect matching.