The threshold for powers of tight Hamilton cycles in random hypergraphs

Abstract

We investigate the occurrence of powers of tight Hamilton cycles in random hypergraphs. For every $r\ge 3$ and $k\ge 1$, we show that there exists a constant $C>0$ such that if $p=p\left(n\right)\ge C{n}^{-1/\left(\begin{array}{c}k+r-2\\ r-1\end{array}\right)}$ then asymptotically almost surely the random hypergraph $H^{\left(r\right)}(n,p)$ contains the kth power of a tight Hamilton cycle. This improves on a result of Parczyk and Person, who proved the same result under the assumption $p=\omega \left(n^{-1/\left(\begin{array}{c}k+r-2\\ r-1\end{array}\right)}\right)$ using a second moment argument.