Color isomorphic even cycles and a related Ramsey problem

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$f_{k}(n,H)$bethesmallestnumberofcolors$c$ 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 $f_k(n,C_4)=\mathrm{\Theta }(n)$ when $k⩾3$ and $n\to \mathrm{\infty }$. 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 $q$.