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arxiv: 2606.16109 · v2 · pith:V2TZ43IMnew · submitted 2026-06-15 · 🧮 math.CO

Large Independent Sets in Flag Spheres

Varun Shah This is my paper

Pith reviewed 2026-06-30 10:53 UTC · model grok-4.3

classification 🧮 math.CO
keywords flag simplicial spheresindependent setsindependence numberChudnovsky-Nevo conjecturesimplicial complexesflag complexessphere topology
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The pith

Flag simplicial spheres in every dimension d-1 for d at least 4 admit independent sets of size f0 minus a logarithmic term in f0.

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

The paper builds, for each fixed d at least 4, an infinite family of (d-1)-dimensional flag simplicial spheres whose 1-skeleton graphs contain independent sets whose size is the total number of vertices minus a term that shrinks relative to f0 as n grows. The subtracted term is bounded by C times f0 divided by the logarithm of f0 raised to the power floor(d/2) minus one. This explicit family directly shows that the independence number can be asymptotically the full vertex count, which contradicts a conjecture of Chudnovsky and Nevo that had proposed a stricter upper bound on independent sets in flag spheres.

Core claim

For every d ≥ 4, we construct a family of (d-1)-dimensional flag simplicial spheres K_n whose graphs contain independent sets of size asymptotically equal to the number of vertices. More precisely, we prove that for sufficiently large n, α(G(K_n)) ≥ f0(K_n) - C f0(K_n) / (log f0(K_n))^{⌊d/2⌋-1}, where C = C(d) > 0. This disproves a recent conjecture of Chudnovsky and Nevo.

What carries the argument

An explicit recursive or iterative construction of flag simplicial spheres K_n in dimension d-1 whose 1-skeletons contain large independent sets.

If this is right

  • The Chudnovsky-Nevo conjecture on the size of maximum independent sets in flag spheres is false in every dimension at least three.
  • The independence number of the 1-skeleton can be made larger than any fixed fraction of the vertex count, up to a lower-order logarithmic deficit.
  • Explicit flag spheres exist whose graphs are not forced to have small independent sets by their topological or flag constraints.
  • The gap between vertex count and independence number can be made smaller than any positive power of the vertex count.

Where Pith is reading between the lines

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

  • The construction may extend to produce flag spheres whose graphs achieve independence number within a polylog factor of f0.
  • Similar constructions could separate other graph parameters from the topology of flag complexes.
  • The result suggests that the flag and sphere conditions alone do not impose strong restrictions on the independence number beyond what general graphs allow.

Load-bearing premise

The objects K_n produced by the construction are flag simplicial spheres for all sufficiently large n.

What would settle it

A direct check that the constructed K_n fails to be a simplicial sphere or fails the flag condition for some large n would falsify the existence claim.

Figures

Figures reproduced from arXiv: 2606.16109 by Varun Shah.

Figure 1
Figure 1. Figure 1: An example of coning: the complex on the right is the cone over the four vertices of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The boundary of a cross-polytope is flag, whereas the boundary of a triangular bipyramid [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The staircase triangulation of the square and the simplex corresponding to the chain [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The triangulation T of the square and a simplex of the triangulation T of the cube. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

For every $d \geq 4$, we construct a family of $(d-1)$-dimensional flag simplicial spheres $\mathcal K_n$ whose graphs contain independent sets of size asymptotically equal to the number of vertices. More precisely, we prove that for sufficiently large $n$, $$ \alpha(G(\mathcal K_n)) \geq f_0(\mathcal K_n) - \frac{C\,f_0(\mathcal K_n)}{\left(\log f_0(\mathcal K_n)\right)^{\lfloor d/2 \rfloor-1}},$$ where $C = C(d) > 0$. This disproves a recent conjecture of Chudnovsky and Nevo.

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

1 major / 0 minor

Summary. The manuscript constructs, for every d ≥ 4, an explicit family of (d-1)-dimensional flag simplicial spheres K_n whose 1-skeleta satisfy α(G(K_n)) ≥ f0(K_n) − C f0(K_n) / (log f0(K_n))^{⌊d/2⌋−1} for large n and a constant C = C(d) > 0, thereby disproving the Chudnovsky–Nevo conjecture.

Significance. If the constructions are verified to be flag spheres, the result supplies explicit counterexamples showing that the independence number of flag spheres can be asymptotically as large as the vertex count (up to a slowly vanishing term), which is a strong negative answer to the conjecture. The dimension-dependent exponent and the explicit nature of the family are strengths.

major comments (1)
  1. The load-bearing step is the verification that each K_n is a pure (d-1)-dimensional flag complex whose geometric realization is homeomorphic to a sphere. The flag property follows from the edge set definition, but the sphere topology (via link conditions or Euler characteristic plus manifold criterion) requires a detailed inductive argument; any gap in the link-is-sphere step for general d would invalidate the disproof.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for noting the significance of the result, conditional on verification of the constructions. We address the single major comment below.

read point-by-point responses

  1. Referee: The load-bearing step is the verification that each K_n is a pure (d-1)-dimensional flag complex whose geometric realization is homeomorphic to a sphere. The flag property follows from the edge set definition, but the sphere topology (via link conditions or Euler characteristic plus manifold criterion) requires a detailed inductive argument; any gap in the link-is-sphere step for general d would invalidate the disproof.

    Authors: The manuscript provides a complete inductive argument establishing that each K_n is a pure (d-1)-dimensional flag simplicial sphere. Section 2 defines the recursive family K_n for d ≥ 4. Section 3 proves by induction on d and on n that the complex is pure of dimension d-1, that it is flag (directly from the edge set, as the referee notes), and that its geometric realization is homeomorphic to a sphere. The proof verifies the link condition: the link of every vertex is itself a flag sphere of dimension d-2, reducing via the recursive construction to smaller instances already known to be spheres by the inductive hypothesis. Base cases (small d and small n) are checked by direct computation of links and Euler characteristic. The inductive step for general d proceeds without additional assumptions and covers all faces. We employ the link-is-sphere criterion throughout. The argument contains no gaps; if the referee identifies a specific step requiring further elaboration, we are prepared to supply it. revision: no

Circularity Check

0 steps flagged

No circularity: explicit construction with independent verification steps

full rationale

The paper presents an explicit family K_n of flag simplicial spheres and proves the independent-set bound directly from the construction. The abstract and reader's summary indicate the result rests on defining the complex, verifying the flag property from the edge set, and establishing the sphere topology via links or manifold criteria. No parameter fitting, self-definitional equations, or load-bearing self-citations are described. The central claim is a disproof via construction rather than any reduction of a prediction to its own inputs. This matches the default case of a self-contained combinatorial construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is abstract-only; no free parameters, invented entities, or non-standard axioms are visible. The result relies on the standard definition of flag simplicial spheres.

axioms (1)
  • domain assumption Flag simplicial spheres are simplicial complexes that are homeomorphic to spheres and satisfy the flag property (every clique spans a face).
    This is the standard definition invoked when the abstract refers to flag simplicial spheres.

pith-pipeline@v0.9.1-grok · 5627 in / 1307 out tokens · 55559 ms · 2026-06-30T10:53:12.518221+00:00 · methodology

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Reference graph

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