Our first main result is a uniform bound, in every dimension $k in mathbb N$, on the topological Turan numbers of $k$-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 $2 le i le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 cup V_2 cup dots cup V_i$. Our second main result is the following estimate for the Turan numbers of degenerate trace-bounded hypergraphs: for all $r ge 2$ and $dinmathbb N$, there is an $alpha_{r,d} ge (5rd)^{1-r}$ such that for any $d$-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).
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