Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs

Abstract

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 $(p,\alpha,\epsilon)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}\alpha n^{k}$ we have $$e(A_1,\dots, A_k)=(1\pm\epsilon)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $\alpha=\alpha(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big(p,\alpha, o(1)\big)$-pseudo-random $n$-vertex hypergraph $H$ with a mild minimum vertex 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, $\big(p,\alpha, o(1)\big)$-pseudo-random $k$-graphs as above contain: $(i)$ a perfect matching if $\alpha=o(p^{k})$ and $(ii)$ a loose Hamilton cycle if $\alpha=o(p^{k-1})$. This extends the works of Lenz--Mubayi, and Lenz--Mubayi--Mycroft who studied the analogous problems in the dense setting.