We prove that for all integers \(\Delta ,r \ge 2\), there is a constant \(C = C(\Delta ,r) >0\) such that the following is true for every sequence \({\mathcal {F}}= \{F_1, F_2, \ldots \}\) of graphs with \(v(F_n) = n\) and \(\Delta (F_n) \le \Delta \), for each \(n \in {\mathbb {N}}\). In every r-edge-coloured \(K_n\), there is a collection of at most C monochromatic copies from \({\mathcal {F}}\) whose vertex-sets partition \(V(K_n)\). This makes progress on a conjecture of Grinshpun and Sárközy.

中文翻译：

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用少量单色有界度图平铺边缘彩色图

我们证明，对于所有整数\(\Delta ,r \ge 2\)，存在一个常数\(C = C(\Delta ,r) >0\) ，使得以下对于每个序列\({\ mathcal {F}}= \{F_1, F_2, \ldots \}\)的图，其中\(v(F_n) = n\)和\(\Delta (F_n) \le \Delta \)，对于每个\( n \in {\mathbb {N}}\)。在每个r边颜色\(K_n\)中，有来自\({\mathcal {F}}\)的最多C 个单色副本的集合，其顶点集分区\(V(K_n)\)。这使得格林什潘和萨科齐的猜想取得了进展。