Ordered Ramsey Theory and Track Representations of Graphs

We study an ordered version of hypergraph Ramsey numbers using 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 vertex order. Moshkovitz and Shapira described its ordered Ramsey number as the solution to an enumeration problem involving higher-order mulidimensional integer partitions. We give a short proof of an equivalent result: the Ramsey number is 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 apply the resulting bounds to study the minimum number of interval graphs whose union is the line graph of the n-vertex complete graph, proving the conjecture of Heldt, Knauer, and Ueckerdt that this number grows with n. In fact, the growth rate is between Ω( log logn log log logn) and O(log log n).