Upper bounds for measures on distal classes

TL;DR

This paper establishes upper bounds on the number of measures on distal Fraïssé classes, providing the first general results and explicit bounds in this area, with applications to classes like s-dimensional permutations.

Contribution

It proves that distal Fraïssé classes with certain automorphism bounds admit finitely many measures and provides explicit upper bounds on their count.

Findings

Finitely many measures exist for distal classes under certain conditions.

Explicit upper bounds are derived, e.g., for s-dimensional permutations.

The bounds grow double-exponentially with the dimension parameter.

Abstract

In recent work, Harman and Snowden introduced a notion of measure on a Fra\"iss\'e class $\mathfrak{F}$, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and so far only a handful of cases have been worked out. In this paper, we obtain some of the first general results on measures. Our main theorem states that if $\mathfrak{F}$ is distal (in the sense of Simon), and there are some bounds on automorphism groups, then $\mathfrak{F}$ admits only finitely many measures; moreover, we give an effective upper bound on their number. For example, if $\mathfrak{F}$ is the class of ``$s$-dimensional permutations'' (finite sets equipped with $s$ total orders), we show that the number of measures is bounded above by approximately $\exp(\exp(s^2 \log{s}))$.