Pseudofiniteness and measurability of the everywhere infinite forest

TL;DR

This paper investigates the properties of infinite-branching and r-regular trees, demonstrating their pseudofiniteness and measurability through ultraproducts of polynomial classes, advancing understanding of their logical and measure-theoretic characteristics.

Contribution

It establishes the pseudofiniteness and measurability of infinite-branching and r-regular trees using ultraproducts, connecting graph theory with model theory and measure theory.

Findings

Both trees are pseudofinite.

They can be realized as ultraproducts of polynomial classes.

They are shown to be generalized measurable.

Abstract

In this paper we study the theories of the infinite-branching tree and the $r$-regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of graphs, and so they are also generalised measurable.