Abstract: We study an ordered version of hypergraph Ramsey numbers for linearly ordered vertex sets, due to Fox, Pach, Sudakov, and Suk. In the $k$-uniform ordered path, the edges are the sets of $k$ consecutive vertices in a linear order. Moshkovitz and Shapira described its ordered Ramsey number in terms of an enumerative problem: it equals $1$ plus the number of elements in the poset obtained by starting with a certain disjoint union of chains and repeatedly taking the poset of down-sets, $k-1$ times. We will describe a proof of this and apply the bounds to study the minimum number of interval graphs whose union is the line graph of the $n$-vertex complete graph. We prove the conjecture of Heldt, Knauer, and Ueckerdt that this value grows with $n$. In fact, the growth rate is between $\Omega({\log\log n}/{\log\log\log n})$ and $O(\log\log n)$. This work is joint with Kevin Milans and Derrick Stolee.