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We prove essentially sharp bounds for Ramsey numbers of ordered hypergraphmatchings, inroduced recently by Dudek, Grytczuk, and Ruci\'{n}ski. Namely, forany $r \ge 2$ and $n \ge 2$, we show that any collection $\mathcal H$ of $n$pairwise disjoint subsets in $\mathbb Z$ of size $r$ contains a subcollectionof size $\lfloor n^{1/(2^r-1)}/2\rfloor$ in which every pair of sets are in thesame relative position with respect to the linear ordering on $\mathbb Z$. Thisimproves previous bounds of Dudek-Grytczuk-Ruci\'nski and ofAnastos-Jin-Kwan-Sudakov and is sharp up to a factor of $2$. For large $r$, weeven obtain such a subcollection of size $\lfloor (1-o(1))\cdotn^{1/(2^r-1)}\rfloor$, which is asymptotically tight (here, the $o(1)$-termtends to zero as $r \to \infty$, regardless of the value of $n$). Furthermore,we prove a multiparameter extension of this result where one wants to find aclique of prescribed size $m_P$ for each relative position pattern $P$. Ourbound is sharp for all choices of parameters $m_P$, up to a constant factordepending on $r$ only. This answers questions of Anastos-Jin-Kwan-Sudakov andof Dudek-Grytczuk-Ruci\'nski.

A sharp Ramsey theorem for ordered hypergraph matchings