Generalizations of the Ruzsa-Szemerédi and rainbow Turán problems for cliques

Abstract

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 $n^r{e}^{-O(\sqrt{\log n})}=n^{r-o(1)}$ 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 $\mathrm{ex}(n,H,$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.

19 pages, 1 figure. Minor changes in Subsection 5.1 compared to the first version

References in corpus (1)

• Generalized rainbow Turán problems

Cited by in corpus (2)

• Rainbow saturation for complete graphs
• The Generalized Turán Problem of Two Intersecting Cliques