Ordered Ramsey numbers of loose paths and matchings

For a kk-uniform hypergraph GG with vertex set {1,…,n}{1,…,n}, the ordered Ramsey number ORt(G)ORt(G) is the least integer NN such that every tt-coloring of the edges of the complete kk-uniform graph on vertex set {1,…,N}{1,…,N} contains a monochromatic copy of GG whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual graph Ramsey numbers. We determine that the ordered Ramsey numbers of loose paths under a monotone order grows as a tower of height two less than the maximum degree in terms of the number of edges. We also extend theorems of Conlon et al. (2015) on the ordered Ramsey numbers of 2-uniform matchings to provide upper bounds on the ordered Ramsey number of kk-uniform matchings under certain orderings.