Cube Width in Poset Theory

• Cube width is a numerical invariant that quantifies the minimal ground set size required for an inclusion representation of a poset.
• It refines the cube height concept by constraining each representing set's size while minimizing the overall union of these sets.
• Sharp bounds, including cw(P) ≤ |P|, characterize when cube width equals the poset size, impacting extremal order and embedding problems.

The cube width of a poset is a numerical invariant that quantifies the minimal combinatorial complexity required to realize an inclusion representation of a given poset using a ground set of bounded size. Specifically, the cube width measures, among all possible inclusion representations of the poset that are “short” (i.e., every representing set has size at most equal to the cube height), the least possible size of the ground set upon which these representing sets are built. This concept extends traditional dimension parameters and arises naturally in the context of extremal set-theoretic and order-theoretic problems.

1. Inclusion Representations and Fundamental Definitions

An inclusion representation of a finite poset $P$ is a family of sets $\mathcal{S} = \{S_x : x \in P\}$ indexed by the elements of $P$, such that for all $x, y \in P$: $$x \leqslant y \text{ in } P \qquad\Longleftrightarrow\qquad S_x \subseteq S_y.$$ A canonical example is $S_x = D_P[x]$, the principal down-set at $x$ in the poset.

The ground set for the inclusion representation is $\bigcup\mathcal{S}$, that is, the union of all the representing sets $S_x$.

Cube height ($\mathrm{ch}(P)$) is defined as the least $h \in \mathbb{N}$ for which there is an inclusion representation of $P$ such that $|S_x| \leq h$ for all $x \in P$: $$\mathrm{ch}(P) = \min\{h : \exists \mathcal{S} \text{ with } |S_x| \leq h \ \forall x\}$$ This parameter measures the minimal levelwise complexity required for the representation.

2. Definition of Cube Width and Main Inequalities

The cube width ($\mathrm{cw}(P)$) of a poset $P$ is the minimum cardinality $w$ of the ground set needed so that there exists an inclusion representation $\mathcal{S}$ with $|\bigcup\mathcal{S}|=w$ and $|S_x| \leq \mathrm{ch}(P)$ for all $x \in P$: $$\mathrm{cw}(P) = \min\{|\bigcup\mathcal{S}| : \mathcal{S} \text{ is an inclusion representation, } |S_x| \leq \mathrm{ch}(P) \ \forall x \in P\}$$ This definition refines the cube height by limiting both the maximum set size and the total ambient dimension needed for the embedding.

An important chain of inequalities is established: $$\mathrm{ch}(P) \leq \mathrm{dim}_2(P) \leq \mathrm{cw}(P) \leq \mathrm{iir}(P) \leq |P|$$ where $\mathrm{dim}_2(P)$ is a generalized “2-dimension” for $P$, and $\mathrm{iir}(P)$ is the maximal size of the ground set in an irreducible inclusion representation.

3. Relationship Between Cube Height and Cube Width

Both parameters are defined via constrained inclusion representations, but their geometric and combinatorial meanings are distinct:

• $\mathrm{ch}(P)$ measures the “local complexity”—the maximal size of sets in any inclusion representation.
• $\mathrm{cw}(P)$ measures the “global complexity”—the minimal size of the union of those sets, under the constraint that no set is larger than the cube height.

Notably, in an antichain (every element incomparable), $\mathrm{ch}(P)=1$ but $\mathrm{cw}(P)=|P|$, since each singleton set must be disjoint.

4. Extremal Results and Bound Theorems

One of the main results of the paper is a resolution to the cube width conjecture: $\mathrm{cw}(P) \leq |P|$ for every finite poset $P$. This means that no matter how intricate the partial order, there always exists a short inclusion representation with ground set no larger than the underlying set of elements in $P$ itself.

The proof leverages an inductive scheme where, by successively removing principal down-sets associated with elements, one can construct inclusion representations whose ground set size never exceeds $|P|$ (cf. Lemma 2.1 and Theorem 2.2).

5. Irreducible Representations and Characterization of Equality Cases

A central topic is the characterization of posets for which the cube width attains the upper bound $\mathrm{cw}(P)=|P|$. Such posets are precisely those where the canonical inclusion representation (down-sets of elements) is, up to isomorphism, the only irreducible representation with ground set size $|P|$. This is formalized via the invariant $\mathrm{iir}(P)$ (maximum ground set size among irreducible inclusion representations), and the equivalence $\mathrm{cw}(P) = |P|$ whenever $\mathrm{iir}(P) = |P|$.

The paper provides concrete structural properties (all testable in polynomial time) that a poset must satisfy for this equality to hold, such as the “No Block is a Chain Property,” the “Two Down Property,” and the “Parallel Pair Property.”

6. Theoretical Consequences, Chain of Parameters, and Related Problems

Cube width fits naturally among other invariants measuring representational efficiency of posets. The chain (restated here for clarity),

$$\mathrm{ch}(P) \leq \mathrm{dim}_2(P) \leq \mathrm{cw}(P) \leq \mathrm{iir}(P) \leq |P|$$

emphasizes the multiscale structure of representation problems for posets.

In particular, cube width bridges the gap between minimal embedding size (cube height), generalized dimension notions, and maximal irreducibility. Extremal results for cube width have applications in induced saturation problems, and related questions in hypercube embeddings and dimension theory.

7. Summary of Main Formulas and Implications

• Definition of cube height:

$$\mathrm{ch}(P) = \min \left\{ h \in \mathbb{N} : \exists \mathcal{S},\ \forall x,\ |S_x| \leq h \right\}$$

• Definition of cube width:

$$\mathrm{cw}(P) = \min \left\{ |\bigcup_x S_x| : \mathcal{S}\ \text{with}\ |S_x| \leq \mathrm{ch}(P) \ \forall x \right\}$$

• Sharp upper bound theorem:

$\mathrm{cw}(P)\leq |P|$

• Chain of inequalities:

$$\mathrm{ch}(P) \leq \mathrm{dim}_2(P) \leq \mathrm{cw}(P) \leq \mathrm{iir}(P) \leq |P|$$

8. Concluding Remarks

The cube width of a poset provides a precise measurement of how efficiently the poset can be embedded, via inclusion representations, in a set-theoretic cube where both the maximal representing set size and the total number of “cube directions” are tightly controlled. The result that cube width never exceeds the size of the poset (with exact characterizations for equality) establishes a fundamental boundary for extremal poset representations, with repercussions for inclusion dimension, saturation problems, and computational complexity of poset invariants (Bastide et al., 1 Oct 2025).