Abstract. Let [Formula: see text] denote the maximum number of edges not contained in any monochromatic copy of [Formula: see text] in a [Formula: see text]-coloring of the edges of [Formula: see text], and let [Formula: see text] denote the Turán number of [Formula: see text]. In place of [Formula: see text] we simply write [Formula: see text]. Keevash and Sudakov proved that [Formula: see text] if [Formula: see text] is an edge-critical graph or [Formula: see text] and asked if this equality holds for any graph [Formula: see text]. All known exact values of this question require [Formula: see text] to contain at least one cycle. In this paper we focus on acyclic graphs and present the following results: (1) We prove [Formula: see text] when [Formula: see text] is a spider or a double broom. (2) We show that a tail in [Formula: see text] is a path [Formula: see text] such that [Formula: see text] is only adjacent to [Formula: see text], and [Formula: see text] is only adjacent to [Formula: see text] in [Formula: see text]. We obtain a tight upper bound for [Formula: see text] when [Formula: see text] is a bipartite graph with a tail. This result provides the first bipartite graphs which answer the question of Keevash and Sudakov in the negative. (3) We answer a question of Liu, Pikhurko, and Sharifzadeh who asked if [Formula: see text] when [Formula: see text] is a tree. We provide an upper bound for [Formula: see text] and show it is tight when [Formula: see text] is prime. This provides a negative answer to their question.

Let $r\ge k\ge 2$ and $K_{r,n}^{(k)}$ denote the complete $n$-balanced $r$-partite $k$-uniform hypergraph, whose vertex set consists of $r$ parts, each has $n$ vertices, and whose edge set contains all the $k$-element subsets with no two vertices from one part. A decomposition of $K_{r,n}^{(k)}$ is a partition of $E(K_{r,n}^{(k)})$. A perfect matching (resp., Hamilton tight cycle) decomposition of $K_{r,n}^{(k)}$ is a decomposition of $K_{r,n}^{(k)}$ into perfect matchings (resp., Hamilton tight cycles). In this paper, we prove that if $k\mid n$ (resp., $2\nmid k$ and $k\mid n$), then $K_{k+1,n}^{(k)}$ (resp., $K_{k+2,n}^{(k)}$) has a perfect matching decomposition. We also prove that for any integer $k\geq 2$, $K_{k+1,n}^{(k)}$ has a Hamilton tight cycle decomposition. In all cases, we use constructive methods involving number theory. In fact, we confirm two conjectures proposed by Zhang, Lu, and Liu [ Appl. Math. Comput., 386 (2020), 125492].