Let Kn(k)\documentclass[12pt]{minimal}
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\begin{document}$${K_n^{(k)}}$$\end{document} be the complete k-uniform hypergraph, k≥3\documentclass[12pt]{minimal}
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\begin{document}$${k\ge3}$$\end{document}, and let ℓ be an integer such that 1 ≤ ℓ ≤ k−1 and k−ℓ divides n. An ℓ-overlapping Hamilton cycle in Kn(k)\documentclass[12pt]{minimal}
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\begin{document}$${K_n^{(k)}}$$\end{document} is a spanning subhypergraph C of Kn(k)\documentclass[12pt]{minimal}
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\begin{document}$${K_n^{(k)}}$$\end{document} with n/(k−ℓ) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of consecutive edges in C intersects in precisely ℓ vertices. An edge-coloring of Kn(k)\documentclass[12pt]{minimal}
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\begin{document}$${K_n^{(k)}}$$\end{document} is (a, r)-bounded if every subset of a vertices of Kn(k)\documentclass[12pt]{minimal}
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\begin{document}$${K_n^{(k)}}$$\end{document} is contained in at most r edges of the same color. In this paper, we refine recent results of the first author, Frieze and Ruciński by proving that there is a constant c = c(k, ℓ) such that every (ℓ,cnk-ℓ)\documentclass[12pt]{minimal}
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\begin{document}$${(\ell, cn^{k-\ell})}$$\end{document} -bounded edge-colored Kn(k)\documentclass[12pt]{minimal}
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\begin{document}$${K_n^{(k)}}$$\end{document} in which no color appears more that cnk-1 times contains a rainbow ℓ-overlapping Hamilton cycle. We also show that there is a constant c′ = c′(k, ℓ) such that every (ℓ, c′nk-ℓ)-bounded edge-colored Kn(k)\documentclass[12pt]{minimal}
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\begin{document}$${K_n^{(k)}}$$\end{document} contains a properly colored ℓ-overlapping Hamilton cycle.
