On quantitative aspects of a canonisation theorem for edge-orderings

For integers (Formula presented.) and (Formula presented.) there are (Formula presented.) canonical orderings of the edges of the complete (Formula presented.) -uniform hypergraph with vertex set (Formula presented.). These are exactly the orderings with the property that any two subsets (Formula presented.) of the same size induce isomorphic suborderings. We study the associated canonisation problem to estimate, given (Formula presented.) and (Formula presented.), the least integer (Formula presented.) such that no matter how the (Formula presented.) -subsets of (Formula presented.) are ordered there always exists an (Formula presented.) -element set (Formula presented.) whose (Formula presented.) -subsets are ordered canonically. For fixed (Formula presented.) we prove lower and upper bounds on these numbers that are (Formula presented.) times iterated exponential in a polynomial of (Formula presented.).