arxiv: 2604.16905 · v1 · submitted 2026-04-18 · 🧮 math.CO

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Lower bounds on the g-numbers of spheres without large missing faces

Isabella Novik , Hailun Zheng

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keywords simplicial spheresg-numbersflag complexesmissing faceslower boundsindependence numberspseudomanifolds

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The pith

Simplicial spheres without large missing faces obey new lower bounds on their g-numbers expressed via vertex count and graph independence numbers.

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

The paper establishes new lower bounds on the g-numbers of simplicial spheres that lack large missing faces. It extends a result of Chudnovsky and Nevo to express these bounds in terms of the independence numbers of the spheres' graphs. This yields that flag (d-1)-spheres and flag normal (d-1)-pseudomanifolds satisfy g2 at least (1/2 minus a vanishing term) times the number of vertices as dimension grows. For simplicial 4-spheres without missing faces of dimension greater than two the bound is g2 at least (2/5) times the vertices minus 6/5. An initial segment of the g-vector is also shown to be a level sequence for such spheres.

Core claim

For simplicial (d-1)-spheres without large missing faces the g-numbers satisfy inequalities extending those known for graphs, in terms of the independence numbers of their 1-skeletons. Consequently flag (d-1)-spheres and flag normal (d-1)-pseudomanifolds obey g2 ≥ (1/2 − δ(d)) f0 where δ(d) tends to zero with d. An initial segment of the g-vector forms a level sequence, producing further inequalities. Simplicial 4-spheres without missing faces of dimension exceeding two satisfy the sharper bound g2 ≥ (2/5) f0 − 6/5.

What carries the argument

Absence of large missing faces, which permits extending the Chudnovsky-Nevo relation between g-numbers and the independence number α of the graph to the setting of spheres and pseudomanifolds.

If this is right

Flag (d-1)-spheres and flag normal (d-1)-pseudomanifolds satisfy g2 ≥ (1/2 − δ(d)) f0 with δ(d) → 0 as d → ∞.
Simplicial 4-spheres without missing faces of dimension >2 satisfy g2 ≥ (2/5) f0 − 6/5.
An initial segment of the g-vector of any simplicial (d-1)-sphere without large missing faces is a level sequence.
The same lower bounds and level-sequence property apply to the broader class of flag normal (d-1)-pseudomanifolds.

Where Pith is reading between the lines

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

The asymptotic 1/2 ratio suggests that high-dimensional flag spheres have 1-skeletons whose independence numbers are comparable to those of random graphs or complete bipartite graphs of balanced parts.
The level-sequence property on the initial g-vector may permit recursive or inductive arguments that bound higher g-numbers once the g2 bound is known.
One could test sharpness by computing g-vectors of explicit high-dimensional flag sphere constructions such as those obtained from stacked polytopes or neighborly complexes.

Load-bearing premise

The Chudnovsky-Nevo theorem on g-numbers extends to simplicial spheres that have no large missing faces.

What would settle it

A sequence of flag (d-1)-spheres in increasing dimension d where the ratio g2/f0 stays bounded below 1/2 by a fixed positive amount independent of d, or a concrete simplicial 4-sphere without missing faces of dimension >2 whose g2 falls below (2/5)f0 − 6/5.

read the original abstract

We establish several new lower bounds on the $g$-numbers of simplicial spheres without large missing faces. For this class of spheres, we derive bounds on the $g$-numbers in terms of the independence numbers of their graphs, extending a result of Chudnovsky and Nevo. As a consequence, we show that flag $(d-1)$-spheres -- and more generally, flag normal $(d-1)$-pseudomanifolds -- satisfy $g_2\geq (1/2-\delta(d))f_0$, where $\delta(d)$ is a function of $d$ with $\delta(d)\to 0$ as $d\to \infty$. We further prove that, for simplicial $(d-1)$-spheres without large missing faces, an initial segment of the $g$-vector forms a level sequence, yielding additional inequalities among the $g$-numbers. Finally, we show that simplicial $4$-spheres without missing faces of dimension greater than two satisfy $g_2\geq \frac{2}{5}f_0 - \frac{6}{5}$.

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.

Desk Editor's Note private letter to a colleague

The paper extends Chudnovsky-Nevo to give lower bounds on g-numbers for spheres without large missing faces, with concrete payoffs for flag spheres and 4-spheres.

read the letter

The main point is that Novik and Zheng adapt the Chudnovsky-Nevo bound, which ties g-numbers to the independence number of the 1-skeleton, so it applies to simplicial spheres that simply lack large missing faces instead of requiring the sphere to be flag. From there they extract g2 at least (1/2 minus a term that shrinks with dimension) times the number of vertices for flag (d-1)-spheres and flag normal pseudomanifolds, plus the explicit inequality g2 at least (2/5)f0 minus 6/5 for simplicial 4-spheres without missing faces of dimension 3 or higher. They also prove that the initial segment of the g-vector is a level sequence for the whole class they consider. These are the new statements; the rest follows standard combinatorial arguments once the extension is in place. The level-sequence claim is a clean extra observation that immediately yields further inequalities among the g_i. The 4-sphere bound is the most immediately usable because it is dimension-specific and does not hide behind an asymptotic error term. The flag-sphere bound is weaker in low dimensions but still improves on what was previously available once d is large enough. The proofs appear to rest on the combinatorial definitions and the prior result rather than on circular reasoning or invented quantities. One limitation is that the error term δ(d) is not pinned down explicitly, so the bound remains loose for moderate d. Another is that the no-large-missing-faces hypothesis, while natural, excludes some spheres that might still be interesting. The paper does not claim sharpness or settle the g-conjecture, and the constants are not shown to be best possible. This is work for people already following f-vector problems for spheres and pseudomanifolds. Anyone tracking progress on the g-conjecture or looking for new linear inequalities on g2 will get direct value from the stated bounds and the level-sequence property. The extension step is the load-bearing part, but the abstract and the logical structure line up without obvious gaps, so the paper is coherent on its own terms. It deserves a serious referee.

Referee Report

1 major / 2 minor

Summary. The paper proves new lower bounds on the g-numbers of simplicial spheres without large missing faces. It extends the Chudnovsky–Nevo theorem to obtain g-number bounds in terms of the independence number of the 1-skeleton for this broader class. As consequences, it shows that flag (d−1)-spheres and flag normal (d−1)-pseudomanifolds satisfy g₂ ≥ (1/2 − δ(d)) f₀ with δ(d) → 0 as d → ∞, that an initial segment of the g-vector is a level sequence for spheres without large missing faces, and that simplicial 4-spheres without missing faces of dimension >2 satisfy g₂ ≥ (2/5) f₀ − 6/5.

Significance. If the central extension holds, the results advance the lower-bound theory for g-vectors of spheres and pseudomanifolds by relaxing the flag hypothesis while retaining strong asymptotic control in high dimensions. The level-sequence property supplies additional combinatorial inequalities that may be useful beyond the g₂ bounds. The concrete 4-sphere inequality is a verifiable special case that could serve as a test for related conjectures.

major comments (1)

[§4, Theorem 4.3] §4, Theorem 4.3 (extension of Chudnovsky–Nevo): the proof invokes the no-large-missing-faces hypothesis to replace the flag condition when bounding the independence number in the link of a vertex; however, the argument appears to require that every vertex link is itself a sphere without large missing faces, which is not explicitly verified for the pseudomanifold case in the statement of the theorem.

minor comments (2)

[§5] The function δ(d) is defined via an optimization over the possible sizes of missing faces; an explicit closed-form expression or at least a table of values for small d would make the asymptotic statement easier to apply.
[Introduction] Notation: the symbol α(G) for the independence number is introduced only in the proof of Theorem 3.1; it should be defined at first use in the introduction or preliminaries.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying this point in the proof of Theorem 4.3. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (extension of Chudnovsky–Nevo): the proof invokes the no-large-missing-faces hypothesis to replace the flag condition when bounding the independence number in the link of a vertex; however, the argument appears to require that every vertex link is itself a sphere without large missing faces, which is not explicitly verified for the pseudomanifold case in the statement of the theorem.

Authors: We agree that an explicit verification is needed to justify applying the argument to normal pseudomanifolds. The no-large-missing-faces property is inherited by vertex links: if τ is a missing face in the link of a vertex v, then τ ∪ {v} is a missing face of the same cardinality in the original complex. This holds for both simplicial spheres and normal pseudomanifolds. In the revised manuscript we will add a short remark (or lemma) immediately preceding the proof of Theorem 4.3 that records this inheritance and notes that the links of normal pseudomanifolds without large missing faces remain normal pseudomanifolds without large missing faces. We will also clarify the statement of Theorem 4.3 to indicate that the same proof yields the stated bound for flag normal (d−1)-pseudomanifolds. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from external extension and combinatorial arguments

full rationale

The paper extends the external Chudnovsky–Nevo result (g-numbers bounded via graph independence number) to the class of simplicial spheres without large missing faces, then obtains the stated consequences for flag spheres and 4-spheres as corollaries. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central inequalities are not equivalent to their inputs by construction but follow from the new extension applied to the given combinatorial hypotheses. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper uses standard definitions of simplicial spheres, missing faces, g-numbers, flag complexes, and normal pseudomanifolds; no new free parameters or invented entities are introduced.

axioms (2)

standard math g-numbers are defined from the f-vector via the standard transformation g_k = f_k − f_{k−1} + … with the usual sign conventions for spheres.
Invoked in every statement about g-numbers.
domain assumption A simplicial complex is a sphere or normal pseudomanifold if it satisfies the appropriate topological or combinatorial manifold conditions.
Used to apply the g-vector theory and the Chudnovsky–Nevo extension.

pith-pipeline@v0.9.0 · 5488 in / 1475 out tokens · 44856 ms · 2026-05-10T06:53:40.604783+00:00 · methodology

discussion (0)

Reference graph

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