SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1999-2008, January 2020.
In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn [Repeated Patterns in Proper Colourings, preprint, https://arxiv.org/abs/2002.00921 (2020)]. 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 Turán 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 $q$.

中文翻译：

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颜色同构偶数周期和一个相关的Ramsey问题

SIAM离散数学杂志，第34卷，第3期，第1999-2008页，2020年1月。
在本文中，我们首先研究了Conlon和Tyomkyn最近提出的一个新的极值问题[适当颜色中的重复模式，预印本，https：//arxiv.org/abs/2002.00921（2020）]。给定一个图形$ H $和一个整数$ k \ geqslant 2 $，令$ f_ {k}（n，H）$为颜色$ c $的最小数量，从而使整个图形$具有适当的边缘着色具有$ c $颜色的K_ {n} $不包含$ H $的顶点不相交颜色同构副本。使用有限域上的多项式的代数性质，我们给出$ K_ {n} $的显式正确边着色，并显示当$ k \时$ f_ {k}（n，C_ {4}）= \ Theta（n）$ geqslant 3 $和$ n \ rightarrow \ infty $。我们在边缘着色中使用的方法可能具有某些独立利益。我们还考虑了一个相关的广义Ramsey问题。对于给定的图形$ G $和$ H，$让$ r（G，H，q）$是$ G $的最少数量的边缘颜色（不一定正确），以使$ H \ subseteq G $的每个副本的边缘一起至少获得$ q $个不同的颜色。建立与指定二部图的Turán数的关系，我们获得了$ r（K_ {n，n}，K_ {s，t}，q）$的一些一般下界，并具有广泛的$ q $范围。