On quantitative aspects of a canonisation theorem for edge-orderings

TL;DR

This paper investigates the minimal size of complete hypergraphs needed to guarantee a canonically ordered subset, providing bounds that grow exponentially with the subset size, advancing understanding of hypergraph orderings.

Contribution

It establishes new lower and upper bounds on the minimal hypergraph size ensuring canonical orderings for subsets, refining the quantitative understanding of hypergraph canonisation.

Findings

Bounds are k times iterated exponential in a polynomial of n.

Provides the first such bounds for the canonisation problem in hypergraphs.

Advances the quantitative theory of hypergraph orderings.

Abstract

For integers $k\ge 2$ and $N\ge 2k+1$ there are $k!2^k$ canonical orderings of the edges of the complete $k$-uniform hypergraph with vertex set $[N] = \{1,2,\dots, N\}$. These are exactly the orderings with the property that any two subsets $A, B\subseteq [N]$ of the same size induce isomorphic suborderings. We study the associated canonisation problem to estimate, given $k$ and $n$, the least integer $N$ such that no matter how the $k$-subsets of $[N]$ are ordered there always exists an $n$-element set $X\subseteq [N]$ whose $k$-subsets are ordered canonically. For fixed $k$ we prove lower and upper bounds on these numbers that are $k$ times iterated exponential in a polynomial of $n$.