Countable Infinite Strongly Dense Poset

• Countable infinite strongly dense posets are ℵ0-sized posets where every interval contains incomparable points, ensuring strong density.
• Their construction using iterative binary tree grafting induces dense interval properties yet deliberately violates the splitting property seen in finite posets.
• This counterexample refines foundational poset theory, prompting reevaluation of how density and maximal antichains interact in infinite combinatorial settings.

A countable infinite strongly dense poset is a partially ordered set (poset) of cardinality $\aleph_0$ (the cardinality of the natural numbers) exhibiting a high degree of interval incomparability. Specifically, it is both dense (every interval contains a point) and strongly dense (every non-empty interval contains a pair of incomparable points), yet—critically in recent research—not every such poset enjoys the splitting property for its maximal antichains. This concept addresses foundational questions in the combinatorial theory of posets, particularly regarding interactions between density-type properties and maximal antichain decompositions.

1. Definitions: Density, Strong Density, and Splitting in Posets

A poset $P = \langle P, \leq \rangle$ is dense if for every distinct $x <_P y$, the open interval $(x, y) = \{z \in P : x <_P z <_P y\}$ is non-empty. Strong density requires a further property: every non-empty interval $(x, y)$ contains a pair of points $z_1, z_2$ such that $z_1 \not\leq_P z_2$ and $z_2 \not\leq_P z_1$ (incomparable). If $|P| = \aleph_0$, $P$ is countably infinite.

A subset $A \subseteq P$ is an antichain if its elements are pairwise incomparable. A maximal antichain is an antichain not strictly contained in any larger antichain. A maximal antichain $A$ splits the poset if it can be partitioned into $D \cup U$, $D \cap U = \emptyset$, satisfying $P = (D) \cup (U)$, where $(D) = \{x: x \leq d \text{ for some } d \in D\}$ and $(U) = \{x: x \geq u \text{ for some } u \in U\}$.

A poset has the splitting property if every maximal antichain splits in this sense (Džamonja, 13 Jan 2026).

2. Historical Context and the Main Theorem

The splitting property for strongly dense posets connects to work by R. Ahlswede, P.L. Erdős, and N. Graham. Finite strongly dense posets always have the splitting property (Ahlswede–Erdős–Graham, 1995). The question of whether this extends to countably infinite strongly dense posets remained open until the construction described in Dzamonja's 2026 paper, which demonstrates that not all countably infinite strongly dense posets possess the splitting property.

The central result, Theorem 2.1 in (Džamonja, 13 Jan 2026), asserts the existence of a countable, infinite, strongly dense poset $\langle C, \leq \rangle$ lacking the splitting property, thus resolving the aforementioned question in the negative.

3. Construction of the Counterexample Poset

The poset $C$ is constructed by iteratively "sprinkling" above each element at every stage a copy of the finite binary tree. Formally, the process is defined as follows:

• Step 0: $$C_0 = \{\langle \emptyset, s, 0 \rangle : s \in {}^{<\omega}2\}$$, ordered by initial segment inclusion $s \unlhd t$ on finite binary sequences.
• Successor Step ($n \to n+1$): Given $(C_n, \leq_n)$, set

$$C_{n+1} = C_n \cup \{ \langle x, s, n+1 \rangle : x \in C_n,\, s \in {}^{<\omega}2 \setminus \{\langle\rangle\} \}.$$

The ordering $\leq_{n+1}$ extends prior orderings and prescribes comparison rules for new and mixed-level elements, ensuring every new level is structured over the previous as a tree of binary trees.

• Completion: $C = \bigcup_n C_n$, with $x \leq y$ defined whenever $x, y$ appear in some $C_n$ and $x \leq_n y$.

This inductive process ensures that, above each point in $C$, copies of $$\left({}^{<\omega}2 \setminus \{\langle\rangle\}\right)$$ are grafted recursively, inducing strong density throughout the construction (Džamonja, 13 Jan 2026).

4. Verification of Strong Density and Failure of Splitting

Strong Density: For any $x < y$ in $C$, by proceeding to a sufficiently high level $n$ with $x$ and $y$ in $C_n$, the inductive structure of $C$ allows the introduction of infinitely many points in $(x, y)$, among which two can always be found that are incomparable. Indeed, every non-empty interval in $C$ contains infinitely many pairwise incomparable points.

Failure of Splitting: A maximal antichain $A$ consisting of two elements at level 1,

$$A = \{ \langle \langle \emptyset, \langle\rangle, 0\rangle,\, \langle 0\rangle, 1\rangle,\; \langle \langle \emptyset, \langle\rangle, 0\rangle,\, \langle 1\rangle, 1\rangle \},$$

cannot be split. Any partition $A = D \cup U$ leaves at least one point in $C$ outside $(D) \cup (U)$. The construction ensures that the children introduced at stage 1 can always be used to exhibit this defect (Džamonja, 13 Jan 2026).

5. Theoretical Implications and Comparisons

The existence of a countable, strongly dense poset without the splitting property demonstrates that strong density alone, even at the minimal infinite cardinal level, fails to imply the splitting property. This contrasts with the finite case, where strong density universally ensures splitting.

Companion research (e.g., Erdős, Discrete Mathematics 163 (1997)) delineates conditions (such as chain-decomposability) under which splitting does hold for infinite posets, but these require structure beyond mere cardinality and density considerations. The "tree-of-trees" counterexample shows that no cardinal or purely density-type property at size $\aleph_0$ suffices for splitting—all such properties are refuted by this explicit construction (Džamonja, 13 Jan 2026).

A plausible implication is that in the hierarchy of combinatorial properties for infinite posets, certain interval-richness conditions do not propagate splitting, highlighting a substantial divergence between finite and infinite poset theory.

6. Extensions and Generalizations

The construction of $C$ can be generalized to higher (possibly uncountable) cardinalities by extending the "grafting" process along longer ordinals in place of $\omega$. Nevertheless, the key phenomenon—that strong density fails to guarantee splitting—appears already at the countable level. The counterexample thus provides a boundary case demarcating where classical density/splitting equivalence for posets breaks down.

In summary, the structure and properties of countable infinite strongly dense posets challenge and refine longstanding conjectures about the interplay between density and antichain decompositions, with ramifications for the study of combinatorial set theory and the foundations of infinite partial orders (Džamonja, 13 Jan 2026).