Summary: Let \(f_k(n,H)\) denote the maximum number of edges not contained in any monochromatic copy of \(H\) in a \(k\)-coloring of the edges of \(K_n\), and let \(\mathrm{ex}(n,H)\) denote the Turán number of \(H\). In place of \(f_2(n,H)\) we simply write \(f(n,H)\). P. Keevash and B. Sudakov [J. Comb. Theory, Ser. B 90, No. 1, 41–53 (2004; Zbl 1033.05042)] proved that \(f(n,H)=\mathrm{ex}(n,H)\) if \(H\) is an edge-critical graph or \(C_4\) and asked if this equality holds for any graph \(H\). All known exact values of this question require \(H\) to contain at least one cycle. In this paper we focus on acyclic graphs and present the following results: (1) We prove \(f(n,H)=\mathrm{ex}(n,H)\) when \(H\) is a spider or a double broom. (2) We show that a tail in \(H\) is a path \(P_3=v_0v_1v_2\) such that \(v_2\) is only adjacent to \(v_1\), and \(v_1\) is only adjacent to \(v_0,v_2\) in \(H\). We obtain a tight upper bound for \(f(n,H)\) when \(H\) is a bipartite graph with a tail. This result provides the first bipartite graphs which answer the question of Keevash and Sudakov [loc. cit.] in the negative. (3) We answer a question of H. Liu et al. [ibid. 135, 16–43 (2019; Zbl 1404.05056)] who asked if \(f_k(n,T)=(k-1)\mathrm{ex}(n,T)\) when \(T\) is a tree. We provide an upper bound for \(f_{2k}(n,P_{2k})\) and show it is tight when \(2k-1\) is prime. This provides a negative answer to their question.