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Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2020-11-19 , DOI: 10.1017/s0963548320000589 W. T. Gowers , Barnabás Janzer
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2020-11-19 , DOI: 10.1017/s0963548320000589 W. T. Gowers , Barnabás Janzer
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 K s such that any K r is contained in at most one K s . 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 K r such that no K r is rainbow.
中文翻译:
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ķ s 这样任何ķ r 最多包含一个ķ s . 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ķ r 这样没有ķ r 是彩虹。
更新日期:2020-11-19
中文翻译:
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