Robust (rainbow) subdivisions and simplicial cycles

Understand

We present several results in extremal graph and hypergraph theory of topological nature.

• First, we show that if $\alpha>0$ and $\ell=\Omega(\frac{1}{\alpha}\log\frac{1}{\alpha})$ is an odd integer, then every graph $G$ with $n$ vertices and at least $n^{1+\alpha}$ edges contains an $\ell$-subdivision of the complete graph $K_t$, where $t=n^{\Theta(\alpha)}$.
• Also, this remains true if in addition the edges of $G$ are properly colored, and one wants to find a rainbow copy of such a subdivision.
• In the sparser regime, we show that properly edge colored graphs on $n$ vertices with average degree $(\log n)^{2+o(1)}$ contain rainbow cycles, while average degree $(\log n)^{6+o(1)}$ guarantees rainbow subdivisions of $K_t$ for any fixed $t$, thus improving recent results of Janzer and Jiang et al., respectively.