The ordered Erdős–Hajnal tower-growth conjecture for tight paths

For integers $2\le k < s < n$ and $2\le t\le {s\choose k}$, let $r_k(s,t;P_n)$ be the minimum $N$ such that every red/blue coloring of the $k$-sets of $[N]$ contains a monochromatic blue copy of the tight path $P_n$ or a set of $s$ vertices inducing at least $t$ red edges. Ordered Erdős–Hajnal tower-growth conjecture. For $3\le t\le k$, there are positive constants $c=c(k,t)$ and $c'=c'(k,t)$ such that

$$\operatorname{twr}_{t-2}(n^c)<r_k(k+1,t;P_n)<\operatorname{twr}_{t-2}(n^{c'}).$$

This conjecture proposes the ordered analogue of the Erdős–Hajnal tower hierarchy for cliques, with tight paths replacing complete hypergraphs; its validity is presented as the paper's main contribution and remains open in the source.