Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of Kr such that no Kr is rainbow.

中文翻译：

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派系的 Ruzsa-Szemerédi 和彩虹 Turán 问题的推广

考虑到 Ruzsa-Szemerédi 问题的自然推广，我们证明对于任何固定的正整数r,s和r<s, 上有图n包含的顶点$n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{ro(1)}$的副本ķs这样任何ķr最多包含一个ķs. 我们还给出了广义彩虹图兰问题 ex (n,H， 彩虹 -F） 什么时候F做完了。特别是，我们回答了 Gerbner、Mészáros、Methuku 和 Palmer 的问题，表明在n顶点与$n^{r-1-o(1)}$的副本ķr这样没有ķr是彩虹。