REVIEW 2 major objections 2 minor
The chromatic number of 3-stable Kneser graphs equals n-3k+3 for large n, and for k=s=3.
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
T0 review · grok-4.5
2026-07-15 02:22 UTC pith:B5WBP4UY
load-bearing objection Solid incremental progress on Meunier’s conjecture for the first odd s>2, but abstract-only so the stability thresholds and topological reduction remain unchecked. the 2 major comments →
The chromatic number of 3-stable Kneser graphs
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The chromatic number of the 3-stable Kneser graph KG of the family of all 3-stable k-subsets of [n] equals n-3k+3 whenever n is large enough (in terms of k), and also equals that value when k=s=3 for every n at least 3k. This confirms Meunier's conjecture in those ranges.
What carries the argument
New Hilton-Milner-type theorems for 3-stable families: structural descriptions of the maximum intersecting families of 3-stable k-sets that are not a star (all sets containing a fixed element). These bounds control the stability threshold that lets the chromatic-number argument go through for large n.
Load-bearing premise
The quantitative thresholds in the new Hilton-Milner theorems for 3-stable sets must kick in at the claimed values of n; if the stability window is larger than stated, the large-n chromatic-number result for s=3 fails.
What would settle it
For a concrete pair (n,k) that the paper claims is already large enough, either produce a proper coloring of the 3-stable Kneser graph with fewer than n-3k+3 colors, or exhibit an intersecting 3-stable family larger than the Hilton-Milner bound the paper uses.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript addresses Meunier’s 2011 conjecture that χ(KG(binom{[n]}{k}_{s-stable})) = n − sk + s for all n ≥ sk. Building on Schrijver’s theorem for s = 2 and on earlier partial results (even s; s ≥ 4 with n large; k = 2), the authors claim a proof for s = 3 whenever n is sufficiently large, and a complete proof for the special case k = s = 3 (n ≥ sk). The argument rests on new Hilton–Milner-type stability theorems for s-stable families that supply the quantitative thresholds needed for the large-n regime; a topological approach toward the general conjecture is also outlined.
Significance. If correct, the result closes a substantial remaining case of Meunier’s conjecture and supplies the first general large-n treatment for an odd stability parameter greater than 1. The new Hilton–Milner theorems for stable families are of independent interest in extremal set theory, and the topological perspective may open a route to the remaining open cases. The claims are falsifiable and sit squarely in the classical combinatorial-topological literature on Kneser graphs.
major comments (2)
- The central large-n claim for s = 3 is stated only as “n large enough.” The load-bearing quantitative thresholds are said to come from the paper’s new Hilton–Milner-type theorems for 3-stable families. Without the body of the manuscript those thresholds, their proofs, and the precise meaning of “large enough” cannot be inspected; any gap or circularity in the stability arguments would collapse the large-n equality. This is an information deficit rather than an identified error, but it prevents verification of the main theorem.
- The abstract asserts that the case k = s = 3 is settled for all n ≥ sk. The same Hilton–Milner machinery (or a separate argument) must underwrite this equality; again the proofs are unavailable for inspection, so the claim cannot be confirmed or refuted from the given text.
minor comments (2)
- The abstract does not indicate whether the topological approach yields any new quantitative bound or is purely conceptual; a one-sentence clarification of its status relative to the combinatorial proofs would help readers.
- Notation for the s-stable Kneser graph is introduced cleanly, but the precise range of parameters for which the new Hilton–Milner statements are proved is left implicit; listing those ranges in the abstract would improve accessibility.
Circularity Check
No circularity: pure combinatorial existence proof; abstract shows no self-definitional, fitted, or load-bearing self-citation reductions.
full rationale
This is an abstract-only review of a pure mathematics paper in combinatorial graph theory. The claimed results are equality statements for chromatic numbers of certain Kneser graphs (χ = n − 3k + 3 for s = 3 and n large enough, and for k = s = 3), obtained via new Hilton–Milner-type theorems for 3-stable families and a topological approach. None of the six circularity patterns can be exhibited: there are no fitted parameters renamed as predictions, no self-definitional identities, no uniqueness theorems imported solely from the authors’ prior work as load-bearing external facts, and no ansatz smuggled via citation. The abstract cites classical results (Schrijver 1978, Meunier’s conjecture) and prior partial resolutions for even s, large n when s ≥ 4, and k = 2; these are independent external benchmarks, not self-citations that force the present equalities by construction. Because the full text is unavailable, deeper citation chains cannot be inspected, but the default for this genre is non-circularity, and nothing in the abstract reduces a claimed derivation to its own inputs. Score 0 is therefore the honest finding.
Axiom & Free-Parameter Ledger
axioms (3)
- domain assumption Standard definition of s-stable subsets and of the Kneser graph on those subsets.
- domain assumption Schrijver’s theorem for 2-stable Kneser graphs and Meunier’s conjecture statement as the target.
- ad hoc to paper Existence of Hilton–Milner-type stability theorems for s-stable families (proved in the paper).
read the original abstract
For an integer $s \ge 2$, a subset $S \subseteq [n]$ is {\em $s$-stable} if $\min \{j - i, n + i - j\}\ge s$ for every $i,j \in S$ with $i<j$. Denote the set of all $s$-stable subsets of size $k$ of $[n]$ by $\binom{[n]}{k}_{s\text{-stable}}$. Schrijver proved in 1978 that whenever $n\ge 2k$, the chromatic number of the Kneser graph $\mathrm{KG}\big( \binom{[n]}{k}_{2\text{-stable}}\big)$ is $n - 2k +2$. Generalizing this result, Meunier conjectured in 2011 that $\chi\left( \mathrm{KG}\big( \binom{[n]}{k}_{s\text{-stable}} \big) \right)= n - sk +s$ for all $n\ge sk$. This conjecture was previously proven for all even $s$, for $s \ge 4$ and large enough $n$, and for $k=2$. We prove the conjecture in the cases $s=3$ and $n$ large enough, or $k=s=3$. To this end, we prove versions of the Hilton-Milner theorem for $s$-stable sets. We also present a topological approach towards Meunier's conjecture.
discussion (0)
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