  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
$ex(n,H)$ denote the Tur\'an number of $H$. In place of $f_2(n,H)$ we simply
write $f(n,H)$. Keevash and Sudakov proved that $f(n,H)=ex(n,H)$ 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 have the following
results:
  (1) We prove $f(n,H)=ex(n,H)$ when $H$ is a spider or a double broom.
  (2) A \emph{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 in the negative.
  (3) Liu, Pikhurko and Sharifzadeh asked if $f_k(n,T)=(k-1)ex(n,T)$ 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.
