The universal exponent conjecture for homeomorphs of 3-graphs

Let $H$ be a fixed $3$-graph. A $3$-graph $G$ contains a homeomorph of $H$ if some subgraph of $G$ is homeomorphic to $H$ as a topological space when both are viewed as $2$-complexes.

Universal exponent conjecture. For each $3$-graph $H$, there exists $C_H>0$ such that every $3$-graph $G$ on $n$ vertices with at least

$$C_H n^{5/2}$$

edges contains a homeomorph of $H$.

The paper proves the analogous statement with exponent $3-1/5$, and the authors expect $5/2$ to be the correct exponent. The conjecture would give a general three-uniform analogue of Mader's theorem for graph subdivisions; its torus specialization is also attributed to Linial.