  In this paper, we first study a new extremal problem recently posed by Conlon
and Tyomkyn~(arXiv: 2002.00921). Given a graph $H$ and an integer $k\geqslant
2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there
exists a proper edge-coloring of the complete graph $K_{n}$ with $c$ colors
containing no $k$ vertex-disjoint color-isomorphic copies of $H$. Using
algebraic properties of polynomials over finite fields, we give an explicit
proper edge-coloring of $K_{n}$ and show that $f_{k}(n, C_{4})=\Theta(n)$ when
$k\geqslant 3$ and $n\rightarrow\infty$. The methods we used in the
edge-coloring may be of some independent interest. We also consider a related
generalized Ramsey problem. For given graphs $G$ and $H,$ let $r(G,H,q)$ be the
minimum number of edge-colors (not necessarily proper) of $G$, such that the
edges of every copy of $H\subseteq G$ together receive at least $q$ distinct
colors. Establishing the relation to the Tur\'{a}n number of specified
bipartite graphs, we obtain some general lower bounds for
$r(K_{n,n},K_{s,t},q)$ with a broad range of $q$.
