Pach–Tardos conjecture for ordered uniform forests

An ordered $r$-uniform forest is an $r$-uniform hypergraph whose vertex set is linearly ordered and whose edges admit an ordering $e_1,e_2,\ldots,e_t$ such that, for every $i\in\{2,3,\ldots,t\}$, there is an $h<i$ with

$$e_i\cap\bigcup_{j<i}e_j\subseteq e_h.$$

The interval chromatic number of an ordered hypergraph is the least number of consecutive intervals in the vertex ordering whose union gives a proper coloring. Let ${\rm ex}_{\rightarrow}(n,F)$ denote the maximum number of edges in an $n$-vertex ordered $r$-uniform hypergraph containing no ordered copy of $F$.

Pach–Tardos conjecture for ordered forests. Let $r\geq 2$. Then for any ordered $r$-uniform forest $F$ with interval chromatic number $r$,

$${\rm ex}_{\rightarrow}(n,F)=O\bigl(n^{r-1}\cdot\operatorname{polylog} n\bigr).$$

This extends the graph conjecture of Pach and Tardos to uniform hypergraphs. The source notes that the corresponding polynomial bound without the polylogarithmic factor is elementary for $r$-uniform forests, but gives no resolution of the strengthened conjecture.