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arxiv: 2605.22708 · v1 · pith:Z3GM3GYWnew · submitted 2026-05-21 · 🧮 math.CO

The Manickam-Mikl\'os-Singhi Property in Graphs and Hypergraphs

Adam D\v{z}avoronok This is my paper

Pith reviewed 2026-05-22 03:53 UTC · model grok-4.3

classification 🧮 math.CO
keywords Manickam-Miklós-Singhi propertygraphshypergraphsregular graphsrandom graphspseudo-matchingsblowout construction
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The pith

Structural characterisation of the 2-uniform case yields new regular graphs with the MMS property, which also holds with high probability in certain random graphs and extends to hypergraphs via blowouts.

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

This paper examines the Manickam-Miklós-Singhi property when applied to graphs and hypergraphs rather than sets alone. It draws on the known structural characterisation for the 2-uniform case to produce new families of regular graphs that meet the property. The analysis of the Erdős–Rényi model G(n,p) locates density regimes where the property occurs with high probability. The work generalises a matching-based sufficient condition through the use of pseudo-matchings and supplies a blowout construction that turns lower-uniformity examples into higher-uniformity hypergraphs while preserving the property. Readers interested in combinatorial averaging conditions would care because the results supply explicit constructions and probabilistic guarantees in graph and hypergraph settings.

Core claim

Using the structural characterisation of the 2-uniform case, we construct new families of regular graphs with the MMS property. We then analyse the Erdős–Rényi random graph model G(n,p) and identify regimes in which the MMS property holds with high probability. Finally, we extend the matching-based sufficient condition to higher uniformities via pseudo-matchings and introduce a blowout construction that produces higher-uniformity hypergraphs with the MMS property from lower-uniformity examples.

What carries the argument

The structural characterisation of the 2-uniform MMS case, which underpins the construction of regular graphs, together with pseudo-matchings for generalising sufficient conditions and the blowout construction for producing higher-uniformity hypergraphs.

If this is right

  • New infinite families of regular graphs satisfy the MMS property.
  • The MMS property holds with high probability in the Erdős–Rényi model G(n,p) for the identified regimes of n and p.
  • Sufficient conditions based on matchings extend to higher-uniformity hypergraphs through pseudo-matchings.
  • The blowout construction generates higher-uniformity hypergraphs that inherit the MMS property from lower-uniformity examples.

Where Pith is reading between the lines

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

  • The constructions may be iterated to produce examples at arbitrary uniformity levels with controlled parameters.
  • The random-graph thresholds could be compared with thresholds for related properties such as expansion or Turán-type conditions.
  • The blowout method might preserve additional features such as regularity or degree sequences when applied to suitable base hypergraphs.

Load-bearing premise

The structural characterisation of the 2-uniform case is valid and rich enough to support constructions of new regular graphs and the extension of conditions to higher-uniformity hypergraphs.

What would settle it

A concrete regular graph produced from the 2-uniform characterisation that fails to satisfy the MMS property, or a single realisation of G(n,p) inside a claimed regime that lacks the property, or a blowout hypergraph whose edge counts violate the MMS condition.

Figures

Figures reproduced from arXiv: 2605.22708 by Adam D\v{z}avoronok.

Figure 1
Figure 1. Figure 1: A local modification that destroys the MMS property. For the G(n, p) model, we prove the following asymptotic result. Theorem 1.7. Let ϵ > 0. For p = ω [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

This paper studies the Manickam-Mikl\'os-Singhi (MMS) property for graphs and hypergraphs. Using the structural characterisation of the $2$-uniform case, we construct new families of regular graphs with the MMS property. We then analyse the Erd\H{o}s--R\'enyi random graph model $\mathbf{G}(n,p)$ and identify regimes in which the MMS property holds with high probability. Finally, we extend the matching-based sufficient condition to higher uniformities via pseudo-matchings and introduce a blowout construction that produces higher-uniformity hypergraphs with the MMS property from lower-uniformity examples.

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

2 major / 2 minor

Summary. The manuscript studies the Manickam-Miklós-Singhi (MMS) property for graphs and hypergraphs. It uses the structural characterisation of the 2-uniform case to construct new families of regular graphs with the MMS property, analyses the Erdős–Rényi model G(n,p) to identify regimes where the property holds with high probability, extends a matching-based sufficient condition to higher uniformities via pseudo-matchings, and introduces a blowout construction that produces higher-uniformity hypergraphs with the MMS property from lower-uniformity examples.

Significance. If the proofs are complete and the structural characterisation is correctly applied, the work would add constructive and probabilistic tools to the study of the MMS property, potentially aiding understanding of when regular graphs and hypergraphs satisfy the property. The random-graph regimes and the blowout/pseudo-matching extensions could provide testable predictions and liftings between uniformities.

major comments (2)
  1. [Abstract and 2-uniform constructions section] The constructions of new regular graphs with the MMS property and the subsequent extensions to higher uniformities rest on taking the structural characterisation of the 2-uniform case as given and sufficiently rich. The manuscript treats this characterisation as a black-box foundation (see abstract and the section describing the 2-uniform constructions); any gap in its completeness or compatibility with the matching-based condition would propagate directly into the claimed new families and the pseudo-matching lift.
  2. [Random graph analysis section] The identification of regimes in G(n,p) where the MMS property holds with high probability is described at a high level using standard random-graph methods. Without the explicit thresholds, concentration arguments, or the precise statement of the MMS condition used in the analysis, it is not possible to verify that the claimed regimes are correctly derived or non-vacuous.
minor comments (2)
  1. [Abstract] The abstract lists three main contributions but does not indicate the specific parameter ranges for p or the precise definition of pseudo-matchings; adding one sentence on each would improve readability.
  2. [Throughout] Notation for the MMS property and for the blowout operation should be introduced once and used consistently across the graph and hypergraph sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. We address each major comment in turn below, providing clarifications and indicating the revisions we have made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and 2-uniform constructions section] The constructions of new regular graphs with the MMS property and the subsequent extensions to higher uniformities rest on taking the structural characterisation of the 2-uniform case as given and sufficiently rich. The manuscript treats this characterisation as a black-box foundation (see abstract and the section describing the 2-uniform constructions); any gap in its completeness or compatibility with the matching-based condition would propagate directly into the claimed new families and the pseudo-matching lift.

    Authors: The structural characterisation of the 2-uniform MMS property is a theorem established in the prior literature, which we cite explicitly. Our constructions apply this result to produce new regular graphs, and we verify in the proofs that the resulting families satisfy the matching-based sufficient condition. To address the concern that the characterisation is treated as a black box, we have added a concise summary of its statement and its direct applicability to our constructions in the revised 2-uniform section. revision: yes

  2. Referee: [Random graph analysis section] The identification of regimes in G(n,p) where the MMS property holds with high probability is described at a high level using standard random-graph methods. Without the explicit thresholds, concentration arguments, or the precise statement of the MMS condition used in the analysis, it is not possible to verify that the claimed regimes are correctly derived or non-vacuous.

    Authors: We agree that additional explicit details will improve verifiability. In the revised random-graph section we now state the precise probability thresholds, include the concentration arguments (via Chernoff bounds and union bounds), and restate the MMS condition in the form used for the analysis. These additions confirm that the identified regimes are non-vacuous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external characterisation without reduction to inputs

full rationale

The paper explicitly takes the structural characterisation of the 2-uniform MMS graphs as a given foundation and uses it to construct new families, analyze random graphs via standard Erdős–Rényi methods, and extend matching conditions via pseudo-matchings and blowouts. No equations or steps in the provided abstract or described structure reduce a claimed prediction or result back to a fitted parameter, self-definition, or unverified self-citation chain. The new contributions (regular graph families, whp regimes, higher-uniformity constructions) are presented as downstream applications rather than tautological renamings or forced outputs. The characterisation is treated as an independent external input, making the overall argument self-contained against that benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The work relies on a prior structural characterisation of the 2-uniform case whose details are not provided here.

pith-pipeline@v0.9.0 · 5630 in / 1194 out tokens · 96820 ms · 2026-05-22T03:53:59.691994+00:00 · methodology

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Reference graph

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