Abstract. Given a graph [Formula: see text] on [Formula: see text] vertices and an assignment of colors to its edges, a set of edges [Formula: see text] is said to be rainbow if edges from [Formula: see text] have pairwise different colors assigned to them. In this paper, we investigate rainbow spanning trees in randomly colored random [Formula: see text] graphs.

We say that a k-uniform hypergraph C is a Hamilton cycle of type $\ell$, for some $1\leq\ell\leq k$, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices, and for every pair of consecutive edges $E_{i-1},E_i$ in C (in the natural ordering of the edges) we have $|E_{i-1}\setminus E_i|=\ell$. We define a class of $(\epsilon,p)$-regular hypergraphs, that includes random hypergraphs, for which we can prove the existence of a decomposition of almost all edges into type $\ell$ Hamilton cycles, where $\ell