pith. sign in

arxiv: 2604.06521 · v1 · submitted 2026-04-07 · 🧮 math.CO

The Exact Saturation Number for the Diamond

Maria-Romina Ivan , Sean Jaffe This is my paper

Pith reviewed 2026-05-10 18:17 UTC · model grok-4.3

classification 🧮 math.CO
keywords saturation numberdiamond posetBoolean latticeinduced subposetmaximal familyextremal set theoryposet avoidance
0
0 comments X

The pith

The smallest maximal diamond-free family of subsets of [n] has size exactly n+1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper settles the saturation number for the diamond poset in the Boolean lattice. A diamond-free family is a collection of sets with no four sets forming an induced diamond under inclusion. The saturation number is the smallest possible size of a maximal such family, one where adding any missing set creates a diamond. A single chain of length n already achieves size n+1 and is maximal and diamond-free. The authors prove that no smaller maximal family exists.

Core claim

Any family of subsets of [n] that is maximal with respect to avoiding an induced copy of the diamond poset Q_2 must contain at least n+1 members. Combined with the fact that a maximal chain has exactly this size and avoids the diamond, the minimal size of a maximal diamond-free family is therefore n+1.

What carries the argument

The diamond (Q_2), the four-element poset with two incomparable middle elements between a unique bottom and top, considered as an induced subposet under set inclusion; the saturation number is the minimal size of a maximal family avoiding an induced copy of this poset.

If this is right

  • The saturation number sat(n, diamond) equals n+1 for every n.
  • No maximal diamond-free family can be smaller than a longest chain.
  • The linear lower bound previously obtained is tight and cannot be improved.
  • Every maximal diamond-free family must intersect every level of the Boolean lattice in a way that blocks smaller constructions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same chain lower bound may hold for other small forbidden posets whose height is two.
  • For larger forbidden posets, the saturation number could still be linear but with a different coefficient.
  • The result suggests checking whether the same exact value holds when the diamond is forbidden only as a weak subposet rather than induced.

Load-bearing premise

That the Boolean lattice is ordered by inclusion and that maximality means adding any missing set immediately creates an induced diamond.

What would settle it

For some n greater than 3, exhibit a diamond-free family of size at most n that is maximal, i.e., every set outside the family forms a diamond with three sets inside the family.

read the original abstract

What is the smallest size of a family of subsets of $[n]$ such that it does not contain an induced copy of $Q_2$ as a poset (known as the \textit{diamond}), but adding a new set creates such a copy? It is easy to see that a maximal chain has this property, and thus the answer is at most $n+1$. Despite the simplicity of the diamond structure, the lower bound stagnated at $\sqrt n$ for quite some time, until recently the authors obtained a linear lower bound. In this paper, we fully solve this question showing that such a family must have size at least $n+1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper determines the exact saturation number sat(2^[n], Q_2) for the diamond poset Q_2 (the induced 4-element poset with two incomparable middle elements) in the Boolean lattice. It proves that every maximal diamond-free family of subsets of [n] has size at least n+1, matching the upper bound achieved by any maximal chain, thereby showing the saturation number is exactly n+1.

Significance. This resolves a long-open question in poset saturation theory. Prior work had improved the lower bound only to linear in n; the exact match to the chain construction is a clean and tight result. The proof relies on direct case analysis of maximality violations using the induced-diamond condition, which is a standard and appropriate technique for this setting.

minor comments (3)
  1. The abstract states the lower bound without indicating the proof method (case analysis on extensions); a brief phrase such as 'via maximality case analysis' would help readers.
  2. Notation for the diamond (Q_2 or the four-set configuration A < B, A < C, B < D, C < D with B || C) is introduced clearly but could be restated once in the main proof section for self-contained reading.
  3. The paper would benefit from an explicit statement of the saturation number definition early in the introduction, including the precise meaning of 'maximal diamond-free'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly captures the main result that the saturation number sat(2^[n], Q_2) equals n+1.

Circularity Check

0 steps flagged

No significant circularity; direct combinatorial proof

full rationale

The paper proves the saturation number is exactly n+1 by exhibiting an explicit upper-bound construction (any maximal chain) and proving the matching lower bound via contradiction: assume a diamond-free family F with |F| < n+1, then derive that maximality must be violated by adding some external set without creating an induced diamond. The argument relies on exhaustive case analysis of the possible inclusion relations that would force a diamond upon extension, using only the standard definition of induced Q2 in the Boolean lattice. No parameters are fitted, no ansatz is smuggled, and the prior linear lower bound (self-cited in the abstract) is not invoked in the new exact proof. The derivation is self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard definitions and properties of the Boolean lattice and induced subposets; no free parameters, new entities, or ad-hoc axioms appear in the abstract.

axioms (2)
  • standard math The Boolean lattice is the power set of [n] ordered by inclusion.
    Fundamental background for defining families of subsets and the diamond poset.
  • domain assumption An induced copy of the diamond (Q2) consists of four sets A < B, A < C, B < D, C < D with B and C incomparable.
    Standard definition of the forbidden configuration in poset theory.

pith-pipeline@v0.9.0 · 5397 in / 1393 out tokens · 88364 ms · 2026-05-10T18:17:52.373034+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Induced poset saturation in the hypergrid

    math.CO 2026-04 unverdicted novelty 7.0

    For every poset P, the induced saturation function sat*([t]^n, P) is either eventually constant or Omega(sqrt(n)) as n grows, with chains constant and unique-twin-cover posets growing.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    Paul Bastide, Carla Groenland, Maria-Romina Ivan, and Tom Johnston,A Polynomial Upper Bound for Poset Saturation, European Journal of Combinatorics (2024)

    work page 2024

  2. [2]

    14 MARIA-ROMINA IVAN AND SEAN JAFFE

    Paul Bastide, Carla Groenland, Hugo Jacob, and Tom Johnston,Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron, Combinatorial Theory4(2024). 14 MARIA-ROMINA IVAN AND SEAN JAFFE

    work page 2024

  3. [3]

    10, 2479–2487

    Michael Ferrara, Bill Kay, Lucas Kramer, Ryan R Martin, Benjamin Reiniger, Heather C Smith, and Eric Sullivan,The Saturation Number of Induced Subposets of the Boolean lattice, Discrete Mathe- matics340(2017), no. 10, 2479–2487

    work page 2017

  4. [4]

    Andrea Freschi, Simón Piga, Maryam Sharifzadeh, and Andrew Treglown,The induced saturation problem for posets, Combinatorial Theory3(2023), no. 3

    work page 2023

  5. [5]

    3, 806–817

    Maria-Romina Ivan,Saturation for the Butterfly Poset, Mathematika66(2020), no. 3, 806–817

    work page 2020

  6. [6]

    ,Minimal Diamond-Saturated Families, Contemporary Mathematics3(2022), no. 2, 81

    work page 2022

  7. [7]

    Maria-Romina Ivan and Sean Jaffe,Gluing Posets and the Dichotomy of Poset Saturation Numbers, arXiv:2503.12223 (2025)

    work page internal anchor Pith review arXiv 2025

  8. [8]

    ,Saturation for Sums of Posets and Antichains, arXiv:2509.10294 (2025)

    work page arXiv 2025

  9. [9]

    ,The Saturation Number for the Diamond is Linear, Bulletin of the London Mathematicsl Society58(2026), e70305

    work page 2026

  10. [10]

    Maria-RominaIvanandNandiWang,Linear Saturation for\via Butterflies, arXiv:2511.08965(2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  11. [11]

    Maria-Romina Ivan,Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

    Balázs Keszegh, Nathan Lemons, Ryan R Martin, Dömötör Pálvölgyi, and Balázs Patkós,Induced and non-induced poset saturation problems, Journal of Combinatorial Theory, Series A184(2021), 105497. Maria-Romina Ivan,Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK. Email add...

    work page 2021