Note on the Splitting Property in Strongly Dense Posets of Size $\aleph_0$

TL;DR

This paper demonstrates that not all countable strongly dense posets possess the splitting property, providing a counterexample to a previously open question in the field.

Contribution

It resolves an open problem by showing the failure of the splitting property in certain countable strongly dense posets.

Findings

Counterexample to the splitting property in countable strongly dense posets

Answers a question posed by Ahlswede, Erdös, and Graham

Clarifies the limitations of the splitting property in infinite posets

Abstract

We show that it is not true that every countable infinite strongly dense poset has the splitting property, so answering a question of R. Ahlswede, P.L. Erd\"os and N. Graham.