Note on the Splitting Property in Strongly Dense Posets of Size $\aleph_0$
Combinatorics
2026-01-14 v1 Logic
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.
Cite
@article{arxiv.2601.08542,
title = {Note on the Splitting Property in Strongly Dense Posets of Size $\aleph_0$},
author = {Mirna Džamonja},
journal= {arXiv preprint arXiv:2601.08542},
year = {2026}
}
Comments
This is an 1998 paper in where we have for the first time used the concept which is called "well-levelled'' orders in our recent preprint arXiv:2512.23003. In the 1998 paper we only used the second level of the well-levelled hierarchy, which is to take a tree and replace every one of its elements by a copy of the original tree. This process is called $T\cdot T$ in arXiv:2512.23003