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

Mirna Džamonja

TL;DR

The paper addresses whether every countably infinite strongly dense poset has the splitting property, answering in the negative with a countable strongly dense counterexample. It constructs the poset by iteratively replacing each node of a binary tree with a copy of the tree and taking the union, yielding a specific maximal antichain that fails to split. This refutes the conjecture and clarifies the limits of the finite case results, situating the finding in the broader discussion of Sperner systems and order theory. The work demonstrates that additional hypotheses are needed for a splitting guarantee in the countable setting.

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. Erdös and N. Graham.

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

TL;DR

The paper addresses whether every countably infinite strongly dense poset has the splitting property, answering in the negative with a countable strongly dense counterexample. It constructs the poset by iteratively replacing each node of a binary tree with a copy of the tree and taking the union, yielding a specific maximal antichain that fails to split. This refutes the conjecture and clarifies the limits of the finite case results, situating the finding in the broader discussion of Sperner systems and order theory. The work demonstrates that additional hypotheses are needed for a splitting guarantee in the countable setting.

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. Erdös and N. Graham.
Paper Structure (2 sections, 1 theorem, 15 equations)

This paper contains 2 sections, 1 theorem, 15 equations.

Table of Contents

  1. Introduction
  2. The example

Key Result

Theorem 2.2

There is a countably infinite strongly dense poset $\langle C,\le\rangle$ which does not have the splitting property. (Moreover, the poset $\langle C,\le\rangle$ has the property described in the statement of Claim one(3) below).

Theorems & Definitions (9)

  • Definition 1.1
  • Definition 1.3
  • Theorem 2.2
  • Definition 2.3
  • Remark 2.4
  • Claim 2.5
  • Claim 2.6
  • Claim 2.7
  • Remark 2.8