We say that a graph F strongly arrows a pair of graphs (G,H) if any colouring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F. The induced Ramsey number, IR(G,H) is defined as the smallest order of a graph that strongly arrows (G,H). We consider the connection between the induced Ramsey number for a pair of two connected graphs IR(G,H) and the induced Ramsey number for multiple copies of these graphs IR(sG,tH), where xG denotes the pairwise vertex-disjoint union of x copies of G. It is easy to see that if F strongly arrow (G,H), then (s+t-1)F strongly arrows (sG, tH). This implies that IR(sG, tH) is at most (s+t-1)IR(G,H). For all known results on induced Ramsey numbers for multiple copies, the inequality above holds as equality. We show that there are infinite classes of graphs for which the inequality above is strict and moreover, IR(sG, tH) could be arbitrarily smaller than (s+t-1)IR(G,H). On the other hand, we provide further examples of classes of graphs for which the inequality above holds as equality.

中文翻译：

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关于多份图的诱导拉姆齐数

我们说图 F 强烈地指向一对图 (G,H)，如果它的边用红色和蓝色进行任何着色导致红色 G 或蓝色 H 作为 F 的诱导子图出现。 诱导拉姆齐数，IR (G,H) 被定义为强烈箭头 (G,H) 的图的最小阶。我们考虑一对两个连通图 IR(G,H) 的诱导拉姆齐数与这些图 IR(sG,tH) 的多个副本的诱导拉姆齐数之间的联系，其中 xG 表示G 的 x 个副本。很容易看出，如果 F 强箭头 (G,H)，那么 (s+t-1)F 强箭头 (sG, tH)。这意味着 IR(sG, tH) 至多是 (s+t-1)IR(G,H)。对于多个副本的诱导拉姆齐数的所有已知结果，上述不等式成立。我们证明有无限类图的上述不等式是严格的，而且 IR(sG, tH) 可以任意小于 (s+t-1)IR(G,H)。另一方面，我们提供了上述不等式为等式的图类的进一步示例。