$f_k(n,H)$ denotes the maximum number of edges not included in the monochrome copy of ~$H$ with $k$ coloring of edges in $K_n$, let be $ex(n,H). $ indicates the Tur\’an number of $H$. Instead of $f_2(n,H)$, 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 that this equation holds for any graph $ I asked if that applies to H$. All known exact values ​​for this question require $H$ to contain at least one cycle. This paper focuses on acyclic graphs and obtains the following results.

    (1) If $H$ is a spider or a broomstick, prove $f(n,H)=ex(n,H)$.

    (2) \emph{tail} of $H$ is the path $P_3=v_0v_1v_2$ where $v_2$ is only adjacent to $v_1$ and $v_1$ is only adjacent to $v_0,v_2$ of $H$ . If $H$ is a bipartite graph with tails, we get a strict upper bound on $f(n,H)$. This result provides the first bipartite graph that answers Keevash and Sudakov’s question negatively.

    (3) Liu, Pikhurko, and Sharifzadeh asked if $f_k(n,T)=(k-1)ex(n,T)$ if $T$ is a tree. We provide an upper bound on $f_{2k}(n,P_{2k})$ and show that it is tight when $2k-1$ is prime. This provides a negative answer to their question.



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