Rainbow Hamilton cycle in hypergraph system

In this paper, we develop a new rainbow Hamilton framework, which is of independent interest, settling the problem proposed by Gupta, Hamann, Müyesser, Parczyk, and Sgueglia when $k=3$, and draw the general conclusion for any $k\ge 3$ as follows. A $k$-graph system $\mathbf{\text{H}}=\{H_i{\}}_{i\in [n]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$, moreover, a $k$-graph $H$ on $V$ is rainbow if $E(H)\subseteq {\bigcup }_{i\in [n]}E(H_i)$ and $\mathrm{\mid }E(H)\cap E(H_i)\mathrm{\mid }\le 1$ for $i\in [n]$. We show that given $\gamma >0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\mathbf{\text{H}}=\{H_i{\}}_{i\in [n]}$ , if ${\delta }_{k-2}(H_i)\ge (5\mathrm{/}9+\gamma )\left(\genfrac{}{}{0px}{}{n}{2}\right)$ for $i\in [n]$ where $k\ge 3$, then there exists a rainbow tight Hamilton cycle. This result implies the conclusion in a single graph, which was proved by Lang and Sanhueza-Matamala [$J\mathrm{.}Lond\mathrm{.}Math\mathrm{.}Soc\mathrm{.},2022$], Polcyn, Reiher, Rödl and Schülke [$J\mathrm{.}Combin\mathrm{.}TheorySer\mathrm{.}B,2021$] independently.