Robust (rainbow) subdivisions and simplicial cycles

Abstract: 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. Furthermore, we consider certain topological notions of cycles in pure simplicial complexes (uniform hypergraphs). We show that if $G$ is a $2$-dimensional pure simplicial complex ($3$-graph) with $n$ $1$-dimensional and at least $n^{1+\alpha}$ 2-dimensional faces, then $G$ contains a triangulation of the cylinder and the M\"obius strip with $O(\frac{1}{\alpha}\log\frac{1}{\alpha})$ vertices. We present generalizations of this for higher dimensional pure simplicial complexes as well. In order to prove these results, we consider certain (properly edge colored) graphs and hypergraphs $G$ with strong expansion. We argue that if one randomly samples the vertices (and colors) of $G$ with not too small probability, then many pairs of vertices are connected by a short path whose vertices (and colors) are from the sampled set, with high probability.