On edge-ordered Ramsey numbers of hypergraphs

For an integer h≥2, an edge-ordered h-uniform hypergraph H^< is an h-uniform hypergraph with a linear ordering of its hyperedges. For an integer q ≥ 1, the edge-ordered Ramsey number Redge(H^<;q) of H^< is the smallest positive integer N such that there exists an edge-ordered h-uniform hypergraph G^< on N vertices such that, for every q-coloring of the hyperedges of G^<, there is a monochromatic copy of H^< in G^< preserving the hyperedge order. In this note, we show that, for every edge-ordered h-uniform hypergraph F^< on n vertices with m≥1 hyperedges, Redge(F^<;q) ≤ Rₕ(F;q)Cm² where F is the underlying unordered hypergraph, Rₕ(F;q) is its standard q-color Ramsey number, and $C=C(h,q)>0$ is a constant. For fixed h≥3 and q≥2, this shows that prescribing the hyperedge order does not require an increase in the tower height of the classical upper bound. It also settles a problem posed by Fox and Li (2020), and for $h=2$ gives the bound 2O(n⁴log n), improving their bounds for every fixed q ≥ 3.