Partition properties of the dense local order and a colored version of Milliken's theorem

TL;DR

This paper investigates the partition properties of the countable homogeneous dense local order, extending Milliken's theorem on trees to derive new finite-dimensional partition results using ideas from the rationals.

Contribution

It introduces a strengthened version of Milliken's theorem applicable to trees, enabling new insights into the partition properties of dense local orders.

Findings

Established finite-dimensional partition properties for the dense local order.

Developed a colored version of Milliken's theorem for trees.

Connected partition calculus of rationals with dense local order properties.

Abstract

We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Milliken's theorem on trees.