An Improved Lower Bound for Diamond-Free Families

Abstract: We construct a diamond-free family in the Boolean lattice whose size is asymptotically larger than the union of two middle layers. Denote the diamond poset by $Q_2$ and let $La(n,Q_2)$ be the maximum size of a family in $2<sup>{[n]}$ containing no weak copy of $Q_2$. We prove $$La(n,Q_2) \ge (c+o(1))\binom{n}{\lfloor n/2\rfloor}$$, where $c \approx 2.147908$. In particular, this disproves the diamond conjecture.

• The paper introduces a novel algebraic construction that produces a lower bound with an asymptotic coefficient of approximately 2.147908, surpassing the union-of-two-layers method.
• It employs a randomized labeling strategy over three consecutive Boolean lattice layers to ensure the exclusion of diamond poset copies.
• The findings refute the longstanding diamond conjecture, opening new avenues in extremal set theory and related combinatorial applications.

An Improved Lower Bound for Diamond-Free Families

Overview and Context

This paper addresses a central open problem in extremal set theory concerning the maximum size of diamond-free families in the Boolean lattice. Denote $Q_2$ as the diamond poset (a 2-dimensional Boolean lattice), and let $La(n, Q_2)$ be the largest cardinality of a family $\mathcal{F} \subseteq 2^{[n]}$ containing no weak copy of $Q_2$. The longstanding "diamond conjecture" posited that $La(n, Q_2) \sim 2 \binom{n}{\lfloor n/2\rfloor}$ as $n\to\infty$; that is, the union of the two largest layers is asymptotically optimal. This work constructs a diamond-free family whose size asymptotically exceeds this value, thereby disproving the conjecture.

Statement of Main Result

The paper proves the following bound: $$La(n,Q_2) \geq \left(2+\frac{29}{135}\prod_{i=1}^\infty (1-4^{-i}) + o(1)\right) \binom{n}{\lfloor n/2\rfloor}$$ with the explicit leading coefficient approximately $2.147908$. This is the first lower bound with coefficient strictly greater than $2$, effectively refuting the diamond conjecture for $Q_2$.

Construction and Methodology

The novel construction operates within the union of three consecutive layers of the Boolean lattice: $La(n, Q_2)$0, $La(n, Q_2)$1, and $La(n, Q_2)$2, where $La(n, Q_2)$3. The method employs a randomized algebraic labelling strategy over the vector space $La(n, Q_2)$4:

• Lower Layer ($La(n, Q_2)$5): Sets for which the assigned vectors span a $La(n, Q_2)$6-dimensional space.
• Middle Layer ($La(n, Q_2)$7): Sets for which the span is at most $La(n, Q_2)$8-dimensional.
• Upper Layer ($La(n, Q_2)$9): Sets for which the span is not $\mathcal{F} \subseteq 2^{[n]}$0-dimensional.

Set inclusion/exclusion is determined by the dimension of the span of labelled vectors. The randomness and algebraic structure ensure that weak $\mathcal{F} \subseteq 2^{[n]}$1-copies (the diamond) are forbidden, as proven in detail.

Probabilistic Analysis

The work rigorously analyzes the limiting densities of sets in each layer satisfying the dimensional constraints. The proof leverages Gaussian binomial coefficients and counts the number of matrices of fixed rank over finite fields. The limiting densities for the three layers are computed to be, respectively, approximately $\mathcal{F} \subseteq 2^{[n]}$2, $\mathcal{F} \subseteq 2^{[n]}$3, and $\mathcal{F} \subseteq 2^{[n]}$4. Summing the contributions from these layers yields the asymptotic leading constant of $\mathcal{F} \subseteq 2^{[n]}$5.

Comparison with Previous Work

Prior constructions, including those based on group-theoretic and hypergraph techniques, could not surpass the $\mathcal{F} \subseteq 2^{[n]}$6 coefficient. Upper bounds have been systematically improved from the trivial $\mathcal{F} \subseteq 2^{[n]}$7 (Erdős’s chain-free result) down through $\mathcal{F} \subseteq 2^{[n]}$8 and $\mathcal{F} \subseteq 2^{[n]}$9 (Griggs, Li, Lu; Axenovich, Manske, Martin; Kramer, Martin, Young; Gro\ss, Methuku, Tompkins). Notably, within three-layer settings, the best upper bound is approximately $Q_2$0 (Balogh, Hu, Lidický, Liu via flag algebras), only $Q_2$1 above the present lower bound.

This construction demonstrates that union of two consecutive layers is not extremal for diamond-free families, even though this heuristic worked for many other posets. The result establishes that such poset-free extremal set theory can exhibit a strict gap beyond the classical "layering" paradigm.

Theoretical and Practical Implications

The result invalidates the conjecture that $Q_2$2 for all finite posets $Q_2$3, where $Q_2$4 is the number of layers that are $Q_2$5-free. This aligns the $Q_2$6 case with recent counterexamples for other posets (e.g., daisies, certain forks) but is remarkable because $Q_2$7 had long been viewed as a borderline case for this phenomenon.

The algebraic construction underscores the tractability and expressive power of vector-space methods for forbidden poset problems, especially in height-2 configurations. This approach might generalize to other poset avoidance problems, especially where prior methods have saturated the combinatorial layering techniques.

From a practical perspective, explicit large diamond-free families have implications for error-correcting codes, combinatorial designs, and database theory, wherever forbidden containment patterns must be avoided.

Future Directions

Potential avenues include:

• Tightening Bounds: The proximity of the lower and upper bounds suggests that even small advances in the construction or analysis techniques could close the gap.
• Field Optimization: The construction is shown to work over $Q_2$8 for any finite field, with $Q_2$9 optimal. Exploring larger or structured fields could yield different phenomena for larger posets.
• Generalization: Extending algebraic spanning-based families to other height-2 posets and higher dimensions.
• Adaptation to Non-uniform Hypergraphs: Analogs for Turán-type and extremal problems can likely benefit from these refined algebraic constructions.

Conclusion

This work establishes a new lower bound for the size of diamond-free families in the Boolean lattice, strictly exceeding the previously conjectured value. The algebraic method reveals that random labelling and span conditions can outperform layer-union strategies, settling a long-standing conjecture and setting a new direction for extremal set theory concerning forbidden subposets ["An Improved Lower Bound for Diamond-Free Families" (2607.09497)].