Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs

We investigate the emergence of spanning structures in sparse pseudo-random k-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A k-uniform hypergraph H on n vertices is called (p, α, e)-pseudo-random if for all not necessarily disjoint sets A1,...,Ak ⊂ V(H) with |A1|···|Ak| > αnk we have e(A1,...,Ak) = (1 ± e)p|A1|···|Ak|. For any linear k-uniform F we provide a bound on α = α(n) in terms of p = p(n) and F, such that (under natural divisibility assumptions on n) any (p, α, o(1))-pseudo-random n-vertex H with a mild minimum degree condition contains an F-factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudo-randomness such as jumbledness or large spectral gap. As a consequence of our results, perfect matchings appear at α = o(pk) while loose Hamilton cycles appear at α = o(pk−1). This extends the works of Lenz-Mubayi, and Lenz-Mubayi-Mycroft who studied the analogous problems in the dense setting.