A sharp Ramsey theorem for ordered hypergraph matchings

Consider two disjoint sets of $r$ integers (or elements from any other totally ordered set). The way the two sets are intertwined can be described by a binary string of size $2r$: for instance the intertwining of the sets $\{0,3,6\}$ and $\{2,4,5\}$ could be described by the pattern $010110$, where the elements of the first sets are represented by 0’s and the elements of the second sets by 1’s.

Given $m$ distjoint $r$-element sets of integers, how many sets can we find that that have pairwise the same intertwining pattern? For $r=2$, Dudek, Grytczuk, and Ruciński completely answered the question by proving that for any $n^3+1$ disjoints sets of integers, $n+1$ of them have pairwise the the same intertwining pattern, while there exists a collection of $n^3$ sets without $n+1$ of them having pairwise the same intertwining pattern.

What about $r\ge 3$?

A recent construction of Anastos, Jin, Kwan and Sudakov gives an upper bound of order $m^{1/(2^r-1)}$ for the problem. The main result of the present paper is a lower bound matching this bound within a factor 2 for all $r\ge 2$, and within a factor tending to 1 for $r\to \infty$.