# A Ramsey Theorem for Multiposets

## TL;DR

This paper generalizes several known Ramsey theorems for finite ordered structures by introducing multiposets with multiple partial and linear orders, proving they have the Ramsey property.

## Contribution

It provides a unified categorical framework to prove the Ramsey property for a broad class of multiposets with fixed extension templates.

## Key findings

- Proves the Ramsey property for classes of finite multiposets.
- Generalizes previous results on posets and linear orders.
- Uses categorical reinterpretation to unify multiple theorems.

## Abstract

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Ne\v{s}et\v{r}il and V. R\"{o}dl claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 M. Soki\'{c} proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of S. Solecki and M. Zhao that the class of all finite posets with several linear extensions has the Ramsey property. Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property.