Transversal Hamilton cycle in hypergraph systems

A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$ and a $k$-graph $H$ on $V$ is said to be $\textbf{H}$-transversal provided that there exists an injection $\varphi: E(H)\rightarrow [m]$ such that $e\in E(H_{\varphi(e)})$ for all $e\in E(H)$. We show that given $k\geq3, \gamma>0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$, if $\delta_{k-1}(H_i)\geq(1/2+\gamma)n$ for each $i\in[n]$, then there exists an $\textbf{H}$-transversal tight Hamilton cycle. This extends the result of Rödl, Ruciński and Szemerédi [Combinatorica, 2008] on single $k$-graphs.