A universal exponent for homeomorphs

Abstract

We prove a uniform bound on the topological Turán number of an arbitrary two-dimensional simplicial complex $S$: any $n$-vertex two-dimensional complex with at least $C_S{n}^{3-1/5}$ facets contains a homeomorphic copy of $S$, where $C_S>0$ is an absolute constant depending on $S$ alone. This result, a two-dimensional analogue of a classical result of Mader for one-dimensional complexes, sheds some light on an old problem of Linial from 2006.

9 pages, submitted