Diamond Subset in Extremal Combinatorics

• Diamond subset is a concept in combinatorics, graph theory, and crystallography that denotes structures excluding the four-element diamond configuration.
• The analysis in Boolean lattices reveals that diamond-free families achieve sizes near 2.25·binom(n, ⌊n/2⌋), highlighting critical extremal bounds and open questions in set theory.
• Construction techniques using abelian groups, Cayley posets, and Markov chains provide practical methods to generate large diamond-free families and isolate forbidden substructures.

A diamond subset, in combinatorics and related fields, refers to a substructure—usually a subset, subposet, or graph subfamily—that is free of a specific configuration known as the diamond. The diamond poset, the diamond graph, and their generalizations arise centrally in extremal set theory, graph theory, and the paper of stacking sequences in crystallography. The concept's precise definition and associated extremal problems vary considerably between these domains, but always center on the exclusion (or precise quantification) of certain four-element combinatorial configurations with specific order or adjacency properties.

1. The Diamond Poset and Diamond-Free Families in the Boolean Lattice

The classical diamond poset, often denoted as $Q_2$ or $D_2$, consists of four sets ordered by $A \subset B, A \subset C, B \subset D, C \subset D$, with $B$ and $C$ incomparable. Within the $n$-dimensional Boolean lattice $\mathcal{B}_n = (2^{[n]}, \subseteq)$, a diamond-free family is a collection of subsets containing no four that realize the diamond poset as an induced (or weak) subposet. This constraint forms the core of the well-studied "diamond problem" in extremal set theory (Kramer et al., 2012, Griggs et al., 2010, Czabarka et al., 2013).

The extremal function $\La(n, D_2)$ represents the largest size of a diamond-free family in $\mathcal{B}_n$. Constructions based on the union of the two middle layers of the lattice yield asymptotically $2 \binom{n}{\lfloor n/2 \rfloor}$ diamond-free sets. The current best universal upper bound is $(2.25 + o(1))\binom{n}{\lfloor n/2 \rfloor}$, proved by Kramer–Martin–Young using a combination of chain-decomposition, flag algebra–style averaging, and combinatorial enumeration (Kramer et al., 2012).

A pivotal analytic tool is the Lubell function: $$\Lambda(\mathcal{F}) = \sum_{F \in \mathcal{F}} \frac{1}{\binom{n}{|F|}},$$ which is the expected number of times a random full chain intersects the family $\mathcal{F}$. Lubell's lemma connects this function to family size. No chain-counting argument can improve the Lubell upper bound beyond 2.25 asymptotically.

2. Generalization to $k$-Diamond Posets and Limiting Densities

The $k$-diamond poset $D_k$ generalizes $D_2$ with $k$ incomparable middle elements: $A < B_1,\dots,B_k < C$. Extremal analysis is conducted via the limiting density

$\pi(D_k) = \lim_{n \to \infty} \frac{\La(n, D_k)}{\binom{n}{\lfloor n/2 \rfloor}},$

when the limit exists. For infinitely many $k$, sharp results have been obtained: for

$m = \lfloor \log_2(k + 2) \rfloor,$

if $k$ falls within specific intervals, then $\pi(D_k) = m$ and the extremal family is unique—the $m$ middle levels of $\mathcal{B}_n$ (Griggs et al., 2010). This pattern generalizes the Erdős–Katona–Sperner principle and shows that the diamond problem sits at the boundary of integer and fractional extremal behaviors.

For the classical $D_2$, the open problem is to determine whether $\pi(D_2) = 2$ holds. All known constructions achieve at most asymptotically $2 \binom{n}{\lfloor n/2 \rfloor}$, and successive upper bounds—currently at $2.25 + o(1)$—have not yet closed this gap.

3. Construction Techniques: Abelian Groups and Markov Chains

General constructions for large diamond-free families have shifted from layer-based approaches to algebraic and probabilistic methods. Czabarka et al. (Czabarka et al., 2013) introduced a technique leveraging Cayley posets built from finite abelian groups. The construction proceeds as follows:

• Form an infinite Cayley poset $P(T,H)$, where $T$ is a finite abelian group of order $m$, and $H \subseteq T$ is an aperiodic generating set.
• Select a finite strongly $D_2$-free subposet $\mathcal{I} \subset P(T,H)$ with $|\mathcal{I}| = \ell$ and no strong chains.
• Weight $[n]$ elements randomly with $H$-values and consider all subsets $A$ of size $\lfloor n/2 \rfloor + i$ whose $H$-sum is prescribed by $\mathcal{I}$.
• Due to Markov chain equidistribution, the expected family size is $({\ell}/{m})\binom{n}{\lfloor n/2 \rfloor}(1+o(1))$.
• By constructing $\mathcal{I}$ with $\ell = 2m$, one obtains asymptotically tight families matching the lower bound for infinitely many $n$.

This group-theoretic method highlights the symmetry constraints needed to avoid diamonds and suggests a structural understanding of extremal or near-extremal diamond-free families.

4. Connections to Graph Theory: The Diamond Graph and Isolation Numbers

In graph theory, the diamond graph is $K_4$ with one edge deleted. The notion of a "diamond-free" subgraph or induced subgraph frequently appears in forbidden subgraph problems.

Isolation numbers further generalize this. Given a connected graph $G$ of order $n \ge 10$, there exists a subset $S \subseteq V(G)$ with $|S| \le n/5$ such that $G[V(G) \setminus N[S]]$ is diamond-free (Yan, 2021). The minimal size of such a set is denoted $\iota(G, B_2)$. This one-fifth bound is sharp and can be achieved by explicit constructions linking diamonds and cycles. The result aligns with parallel bounds for clique and cycle isolation numbers and provides a polynomial-time (even linear-time) algorithm for constructing such subset isolators.

5. Diamond Subset Structures in Crystallography: Stacking-Disordered Diamond

In crystallography, the term "diamond subset" arises in the context of stacking sequences. Both cubic ($Fd\bar{3}m$) and hexagonal (lonsdaleite, $P6_3/mmc$) diamond structures are built from puckered, graphene-like layers of sp$^3$-bonded carbon. The pure cubic system is an ...ABC... stacking and the hexagonal forms an ...AB... stacking. However, most real specimens—especially so-called hexagonal diamond or lonsdaleite—are best described as stacking-disordered diamond, i.e., arbitrary sequences or "subsets" of cubic and hexagonal stacking events (Salzmann et al., 2015).

Hexagonality, denoted $\phi_h$, quantitatively describes the fraction of hexagonal stacking: $$\phi_h = \frac{\text{number of hexagonal stacking events}}{\text{total number of stacking events}}.$$ Experiments utilizing X-ray diffraction and DIFFaX modeling show that even the best samples do not exceed $\phi_h \approx 0.6$; perfect hexagonal stacking remains experimentally unattained. Stacking disorder thus defines a "subset" of stacking sequences that avoid either pure cubic or pure hexagonal continuity.

6. Open Problems and Broader Implications

The intrinsic structure of diamond-free subsets—across posets, graphs, or stacking sequences—poses multiple unresolved problems:

• In extremal set theory, the existence and value of $\pi(D_2)$ remain open. Determining if $\pi(D_2) = 2$ would resolve a major conjecture, potentially via group-based constructions or novel combinatorial partitions (Griggs et al., 2010, Czabarka et al., 2013).
• The universality of isolation number bounds and their extension to other forbidden subgraphs poses ongoing challenges, both in bounding constants and in characterizing extremal configurations (Yan, 2021).
• In materials science, synthesizing diamond specimens that attain significantly higher hexagonality—realizing true lonsdaleite—remains experimentally elusive, with significant implications for predicted mechanical and electronic properties (Salzmann et al., 2015).

A common thread among these questions is the search for maximal structures (or maximal avoidance) under a diamond-type exclusion constraint, whether in algebraic, combinatorial, graph-theoretic, or physical stacking contexts.

7. Summary Table: Diamond Subset Concepts Across Domains

Context	Diamond Subset Definition	Main Extremal Bound / Characterization
Boolean lattice	Subset family omitting $D_2$ poset	$2 \leq \pi(D_2) \leq 2.25$
Graph theory	Vertex set isolating all diamond subgraphs	Isolation number $\leq n/5$
Stacking disorder	Sequence subset avoiding pure cubic or hexagonal wholeness	Hexagonality $\phi_h \lesssim 0.6$

This encapsulates the central role of diamond subsets as objects of combinatorial exclusion, structural quantification, and ongoing foundational inquiry across multiple mathematical and physical disciplines.