Simplicial homeomorphs and trace-bounded hypergraphs

Our first main result is a uniform bound, in every dimension $k \in \mathbb N$,onthetopologicalTur\acute{a}nnumbersof$k$-dimensional simplicial complexes: for each $k\in \mathbb{N}$, there is a ${\lambda }_k\ge k^{-2k^2}$ such that for any $k$-complex $\mathcal{S}$, every $k$-complex on $n\ge n_0(\mathcal{S})$ vertices with at least $n^{k+1-{\lambda }_k}$ facets contains a homeomorphic copy of $\mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of ${\lambda }_1$ is a result of Mader from 1967, and the existence of ${\lambda }_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1,V_2,\dots ,V_r$ is said to be $d$-trace-bounded if for each $2 \le i \le r$,alltheverticesof$V_i$havedegreeatmost$d$inthetraceof$H$on$V_1 \cup V_2 \cup \dots \cup V_i$. Our second main result is the following estimate for the Tur'an numbers of degenerate trace-bounded hypergraphs: for all $r\ge 2$ and $d\in \mathbb{N}$, there is an $\alpha_{r,d} \ge (5rd)^{1-r}$suchthatforany$d$-trace-bounded$r$-partite$r$-graph$H$, every $r$-graph on $n\ge n_0(H)$ vertices with at least $n^{r-{\alpha }_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).