Sharp Bounds on the Independence Number of Simplicial Spheres

Ethan X. Fang; Hailun Zheng; Jes\'us A. De Loera; Junwei Lu; Tingzhou Wei

Sharp Bounds on the Independence Number of Simplicial Spheres

T0 review · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read For every fixed dimension $d\ge4$, the largest independent set of vertices in a simplicial $(d-1)$-sphere on $n$ vertices has size $n-\Theta(n^{1/\lfloor d/2\rfloor})$; for flag spheres the same order holds in dimensions 4 and 5.

desk verdict The beta(d,n) asymptotic is a real step forward; the flag-sphere results are strong too, but the load-bearing identity in Lemma 3.3 needs to be written out before I fully trust it. read the letter →

arxiv 2608.05568 v1 pith:52GNSMJX submitted 2026-08-06 math.CO

Jesús A. De Loera , Ethan X. Fang , Junwei Lu , Tingzhou Wei , Hailun Zheng This is my paper · ORCID

classification math.CO MSC 05E4552B70

keywords independencenumbersimplicialspheresflagBierg-vectorDehn-SommervillerelationsstellarsubdivisionKruskal-Katonatheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper determines, up to a constant factor, how large an independent set of vertices can be in the graph of a simplicial sphere. For every fixed dimension $d\ge4$, it proves a sphere on $n$ vertices can have an independent set of size $n-\Theta(n^{1/\lfloor d/2\rfloor})$, and no more. For flag spheres, the clique complexes of their graphs, the paper proves a lower bound $n-4\sqrt n+O(1)$ in every dimension at least 4, and shows that in dimensions 4 and 5 this is the correct order, so the flag condition does not change the asymptotic answer there. It also proves that Bier spheres, a very large family built from arbitrary simplicial complexes, never have independence number larger than $\lfloor n/2\rfloor$, with equality attainable for every $n\ge3$.

What carries the argument

The load-bearing object is the face-number package of a simplicial sphere: the $h$-vector, the derived $g$-vector, and the Dehn-Sommerville relations expressing high-dimensional face counts in terms of low-dimensional ones. The upper bound on $\beta(d,n)$ runs through an inequality comparing $f_{d-2}(\Delta\setminus I)$ with the $g$-numbers of vertex links; if the independent set $I$ is too large, the inequality forces $\Delta\setminus I$ to keep about $n^{1/\lfloor d/2\rfloor}$ vertices. The flag lower bound uses a different mechanism: take the join of two cycles (plus octahedra in higher dimensions), choose maximum independent sets in each cycle, and stellar-subdivide every edge between the two sets; edge subdivision preserves flagness and creates new vertices that are pairwise nonadjacent.

What would settle it

For some fixed $d\ge4$, build a family of simplicial $(d-1)$-spheres on $n$ vertices whose independence number is $n-o(n^{1/\lfloor d/2\rfloor})$, for instance with deficit $n^{1/\lfloor d/2\rfloor}/\log n$; any such family refutes Theorem 1.2. For dimensions $4$ and $5$, a flag sphere with independence number larger than $n-\sqrt{12n}+O(1)$ for infinitely many $n$ would refute Theorem 4.7.

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Extended reading notes

Core claim

The central result is Theorem 1.2: for each fixed $d\ge4$, $\beta(d,n)=n-\Theta(n^{1/\lfloor d/2\rfloor})$, where $\beta(d,n)$ is the maximum size of an independent set over all simplicial $(d-1)$-spheres on $n$ vertices. The lower bound comes from stellar-subdividing every facet of the cyclic $d$-polytope, and the matching upper bound follows from face-number inequalities combined with the Kruskal-Katona theorem. For flag spheres, Theorem 1.3 gives $\alpha(d,n)\ge n-4\sqrt n+O(1)$ for all $d\ge4$, achieved by edge-subdividing a join of two cycles, and for $d=4,5$ gives $\alpha(d,n)\le n-\sqrt{12n}+O(1)$, so $\alpha(4,n)=\alpha(5,n)=n-\Theta(\sqrt n)$. Finally, Theorem 1.4 shows every Bier sphere with $n$ actual vertices has independence number at most $\lfloor n/2\rfloor$, and this bound is sharp for every $n\ge3$.

Load-bearing premise

The upper bound for $\beta(d,n)$ rests on the assumption that every simplicial sphere has a nonnegative $g$-vector, a package of alternating face-count differences, and that every vertex link has at least $d$ facets; if either of these gave way, the contradiction argument in Lemma 3.3 would collapse.

Editorial extensions

If this is right

For fixed $d\ge4$, the extremal independence deficit of a simplicial $(d-1)$-sphere on $n$ vertices is a power of $n$: $n-\beta(d,n)=\Theta(n^{1/\lfloor d/2\rfloor})$, so the deficit grows as $n^{1/m}$ with $m=\lfloor d/2\rfloor$.
The flag construction yields explicit spheres with independence number exactly $t^2$ on $4t+2(d-4)$ vertices, realizing the lower bound $n-4\sqrt n+O(1)$.
In dimensions $d=4$ and $5$, the flag condition does not change the asymptotic order of the maximum independence number: both flag and general spheres have deficit $\Theta(\sqrt n)$, with constants between $\sqrt{12}$ and $4$.
Bier spheres are never extremal for independence: their independence number is at most $\lfloor n/2\rfloor$, far below the $n-o(n)$ achieved by general spheres in dimension at least 4.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because a size-$r$ independent set makes every induced subcomplex on its subsets disconnected, the face ring of an extremal sphere should carry long linear strands in its minimal free resolution; one could test whether the Betti tables of the constructed spheres visibly reflect this signature.
The formula expressing the independence number of a Bier sphere through the independence numbers of the input complex and its Alexander dual makes Bier spheres a convenient laboratory for engineering spheres with prescribed small independence numbers; they could be used to probe the open $d\ge6$ gap.
The flag extremal examples are joins of two cycles, the same complexes that maximize face numbers of flag 3-spheres; one might conjecture that in higher even dimensions the extremal flag spheres are also joins of low-dimensional flag spheres, although the paper does not prove this.
For $d=6$, the concrete next step is to compute the independence numbers of the $\Gamma_{6,s}$ family and compare them with the best upper bound for flag 6-spheres, since this is where the lower-bound order $n-O(\sqrt n)$ and the inherited upper-bound order $n-\Omega(n^{1/3})$ first disagree.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a careful read. The headline is Theorem 1.2: for every fixed d at least 4, beta(d,n) = n - Theta(n^{1/floor(d/2)}), matching the stellar-subdivision lower bound. The flag sphere results are also a real step: alpha(d,n) at least n - 4 sqrt(n) + O(1) for all d at least 4, and exact order for d=4,5, which supersedes Shah's recent disproof of the Chudnovsky–Nevo conjecture. The Bier sphere bound alpha at most floor(n/2) is clean and sharp. The paper is honest about the remaining gap for d at least 6.

What the paper does well: it combines Dehn–Sommerville relations, g-nonnegativity, and Kruskal–Katona in a coherent way. The lower-bound construction via edge subdivisions preserving flagness is natural given Zheng's flag upper bound theorem. The proofs are mostly checkable; the counting arguments are detailed, and the Bier sphere section is self-contained with an explicit graph description.

The main soft spot is the identity in Lemma 3.3, displayed in (3.3)–(3.4). It rewrites a face-number combination of each vertex link as (1/2) f_{d-2}(L) plus weighted g-terms, but the text only says 'Here we used (3.1) and Lemma 2.3' without showing the algebra. This identity is load-bearing: a single wrong binomial coefficient would break the lower bound (d/2)|I| and the Kruskal–Katona contradiction. I checked small cases and believe it is correct, but 'believe' is not 'verified'. A referee should ask for the derivation or a machine check. The Appendix certificate for the 21-vertex flag 3-sphere is described but not embedded in the text; it is an ancillary file, so a referee will need to run it. That is acceptable, but the paper should make the file's location explicit.

Citation pattern: the only author self-citation is Zheng's flag upper bound theorem, used as motivation, not as a premise. External inputs are standard published results. No circularity, no fitted parameters.

This deserves a serious referee. The main claims are important and likely correct, but the Lemma 3.3 identity should be fully justified. I would send it to peer review, not desk reject, and would expect acceptance after a revision that spells out that step.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the β and α bounds are obtained from explicit constructions plus independent face-number theorems; the one Zheng self-citation is motivational and non-load-bearing.

full rationale

The paper's derivation chain is self-contained in the sense that matters. The lower bound on β(d,n) in Lemma 3.2 is an explicit stellar-subdivision construction based on the Chudnovsky–Nevo example, and the upper bound in Lemma 3.3 derives from Dehn–Sommerville relations, the nonnegativity of g-vectors for simplicial spheres, the Upper Bound Theorem, and Kruskal–Katona. None of these inputs contains the target asymptotic order n−Θ(n^{1/floor(d/2)}). The abbreviated simplification in (3.3)–(3.4) is asserted with the sentence 'Here we used (3.1) and Lemma 2.3,' which is a proof-completeness gap rather than a circular step: the identity does not fit constants from the conclusion or assume β(d,n), and the calculation reduces to independent algebraic facts. The flag-sphere results are supported by the explicit Γ_{d,s} construction and by Dehn–Sommerville and Davis–Okun inequalities; the cited result of Zheng [20] is used only as motivation for the join-of-cycles construction and is not a load-bearing premise. The Bier-sphere analysis is a direct computation from the definition of the Bier complex. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to force a choice, and no known result is merely renamed. Thus the paper exhibits no circularity of the kind that would require a corrective score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are introduced; the results are rigorous asymptotic theorems. The main external loads are Dehn-Sommerville, the Upper Bound Theorem, g-vector nonnegativity, Davis-Okun, and Bier's theorem. No new entities are postulated. The construction Gamma_{d,s} is a new object, not a new postulated entity, and its extremal property is proved in the paper.

assumptions (7)

standard math Dehn-Sommerville relations h_i = h_{d-i} for simplicial (d-1)-spheres.
Used in Lemma 2.1 to express f_{d-2} in terms of h-numbers and in Lemma 4.6 for d=4,5.
standard math Upper Bound Theorem and nonnegativity of the g-vector for simplicial spheres (Lemma 2.3).
Invoked in Lemma 3.3; g-nonnegativity is the deep generalized lower bound theorem (Adiprasito; Karu-Xiao).
standard math Kruskal-Katona bound: a simplicial complex on r vertices has at most C(r,j) (j-1)-faces.
Used in Lemma 3.3 to show if the complement of the independent set is o(n^{1/floor(d/2)}), then low face counts are o(n).
standard math Davis-Okun inequality f1 - 5 f0 + 16 >= 0 for flag 3-spheres, and vertex links of flag 4-spheres are flag 3-spheres with at least 8 vertices.
Used in Lemma 4.6 (d=5) and Theorem 4.7 to control f2(A) and g2 of links.
standard math Edge stellar subdivision preserves flagness and the PL sphere property.
Used in Section 4.1 to construct flag spheres Gamma_{d,s}; cited from Lutz-Nevo.
domain assumption In a flag 2-sphere, f1 = 3 f0 - 6; every edge link in a flag 3-sphere is a cycle of length at least 4.
Used in Lemma 4.6 d=4 to bound f1(lk(u,A)) for a flag 3-sphere Delta; these are structural properties of flag triangulations.
standard math Bier's theorem: the deleted join Bier(K) is a combinatorial (m-2)-sphere.
Basis for all results in Section 5; cited from Bjorner et al.

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Cite this review

Pith. "Pith review of Sharp Bounds on the Independence Number of Simplicial Spheres." pith.science (2026). https://pith.science/paper/52GNSMJX

@misc{pith2026260805568,
  author       = {Pith},
  title        = {Pith review of: Sharp Bounds on the Independence Number of Simplicial Spheres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/52GNSMJX}},
  note         = {Machine review of arXiv:2608.05568}
}

abstract

We study the maximum size of an independent set in the graph of a simplicial sphere. Let $\beta(d,n)$ denote this maximum over all simplicial $(d-1)$-spheres on $n$ vertices, and let $\alpha(d,n)$ denote the maximum restricted to flag $(d-1)$-spheres. For every fixed $d\geq4$, we prove $\beta(d,n)=n-\Theta(n^{1/\lfloor d/2\rfloor})$. For flag spheres, we show $\alpha(d,n)\geq n-4\sqrt n+O(1)$ for all $d\geq4$ and determine the correct asymptotic order $\alpha(d,n)=n-\Theta(\sqrt n)$ for dimensions $d=4,5$. We also investigate the independence sets of Bier spheres and show that, in contrast to our other results, for this very large family of spheres, the independence number cannot be larger than $\left\lfloor\frac{n}{2}\right\rfloor.$

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Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Upper bound theorem for odd-dimensional flag triangulations of manifolds.Mathematika, 62(3):909–928, 2016

Micha l Adamaszek and Jan Hladk´ y. Upper bound theorem for odd-dimensional flag triangulations of manifolds.Mathematika, 62(3):909–928, 2016

work page 2016

[2] [2]

Combinatorial Lefschetz theorems beyond positivity, 2018

Karim Adiprasito. Combinatorial Lefschetz theorems beyond positivity, 2018. arXiv:1812.10454v4

arXiv 2018

[3] [3]

Athanasiadis

Christos A. Athanasiadis. Some combinatorial properties of flag simplicial pseudomanifolds and spheres. Arkiv f¨ or Matematik, 49(1):17–29, 2011

work page 2011

[4] [4]

Topological methods

Anders Bj¨ orner. Topological methods. In Ronald L. Graham, Martin Gr¨ otschel, and L´ aszl´ o Lov´ asz, editors,Handbook of Combinatorics, volume 2, pages 1819–1872. North-Holland, Amsterdam, 1995

work page 1995

[5] [5]

Anders Bj¨ orner, Andreas Paffenholz, Jonas Sj¨ ostrand, and G¨ unter M. Ziegler. Bier spheres and posets. Discrete & Computational Geometry, 34(1):71–86, 2005

work page 2005

[6] [6]

Stable sets in flag spheres.European Journal of Combinatorics, 110:Paper No

Maria Chudnovsky and Eran Nevo. Stable sets in flag spheres.European Journal of Combinatorics, 110:Paper No. 103699, 9 pp., 2023

work page 2023

[7] [7]

Davis and Boris Okun

Michael W. Davis and Boris Okun. Vanishing theorems and conjectures for theℓ 2-homology of right- angled Coxeter groups.Geometry & Topology, 5:7–74, 2001

work page 2001

[8] [8]

The topology of the independence complex.European Journal of Combinatorics, 27(6):906–923, 2006

Richard Ehrenborg and G´ abor Hetyei. The topology of the independence complex.European Journal of Combinatorics, 27(6):906–923, 2006

work page 2006