The tight tree Ramsey growth conjecture for non-trivial tight hypergraph trees

Let $r\text{-graph}$ mean an $r$-uniform hypergraph. A tight $r$-tree is an $r$-graph whose edges can be ordered as $e_1,\dots,e_t$ so that for every $i\geq 2$ there exist $v\in e_i$ and $1\leq s\leq i-1$ with $v\notin\bigcup_{j=1}^{i-1}e_j$ and $e_i\setminus\{v\}\subset e_s$. A tight $r$-tree is non-trivial if no vertex is contained in all of its edges. For an $r$-graph $F$, let $R(F,n)$ be the least integer $N$ such that every $F$-free $N$-vertex $r$-graph has an independent set of size $n$.

Tight tree Ramsey growth conjecture. For $r\geq 5$, if $T$ is a non-trivial tight $r$-tree, then there exist constants $c_1,c_2>0$ such that, for every positive integer $n$,

$$c_1n^{r-1}\leq R(T,n)\leq c_2n^{r-1}.$$

The paper proves this order of growth for $r=3$ and $r=4$. The conjecture extends that result to all higher uniformities; the source notes that natural generalizations of the known constructions already fail for $r=5$, so the lower bound remains the main obstacle.