REVIEW 4 minor 1 cited by
Minimum degree in simplicial complexes
T0 review · 0 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read For every $d\ge m\ge 1$, the paper proves $\alpha(2^d-m)=\frac{2^{d+1}-m}{d+1}$, and it also settles $\alpha(11)=\frac{53}{10}$.
desk verdict Resolves a named conjecture and determines alpha on a full block; proof is long but sound, with only peripheral unsupported claims to clean up. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The proof works with \u201cmountains\u201d, i.e., $N_{\ge 2}$-complexes: set systems closed under taking subsets of size at least two. On a fixed vertex set, several mountains are considered together, and the key structural object is a \u201cconglomerate\u201d, a set $K$ of size $d+1$ whose subsets are almost all present in the first mountain. The load-bearing engine is Claim 5.2: every inclusion-maximal set $M$ of the first mountain contains at least three vertices whose degree inside that mountain is at most $2^d-m$. This assertion rules out sets larger than $d$, fixes the size of conglomerates, and drives all later degree estimates. Around it, a weighted version of the Kruskal-Katona theorem (Lemma 3.8) provides the local lower bounds comparing arbitrary mountains to initial segments of sets.
What would settle it
Search exhaustively for a simplicial complex with minimum degree at least $13$ and $|S|/|V|\le 28/5$; Theorem 1.1 with $d=4$, $m=4$ says no such complex exists, so a single example would refute the claimed threshold.
Extended reading notes
Core claim
The central claim is Theorem 1.1: for all $d\ge m\ge 1$, $\alpha(2^d-m)=\frac{2^{d+1}-m}{d+1}$, where $\alpha(d)$ is the largest real number with the property that every simplicial complex $S$ satisfying $0<|S|\le \alpha(d)|V(S)|$ has some vertex of degree at most $d$. The companion Theorem 1.2 evaluates $\alpha(11)=\frac{53}{10}$. Upper bounds come from explicit constructions that delete a few large sets from a power set on $d+1$ vertices, while the lower bounds form the main body of the paper: if a complex has minimum degree greater than $2^d-m$, it must have more than $\frac{2^{d+1}-m}{d+1}$ edges per vertex. The proof achieves this by weighting vertices, decomposing the complex into \u201cconglomerates\u201d, and applying a weighted form of the Kruskal-Katona theorem locally.
Load-bearing premise
The proof rests on Claim 5.2, the assertion that every inclusion-maximal set in the main mountain contains at least three vertices of degree at most $2^d-m$; if a boundary case allowed only two such vertices, the conglomerate decomposition and the lower bound would collapse.
Editorial extensions
If this is right
- For every $d$, the exact value of $\alpha$ is now known throughout the interval $[2^d-d,\,2^d]$.
- Any simplicial complex with minimum degree at least $2^d-m+1$ must have more than $\frac{2^{d+1}-m}{d+1}$ edges per vertex.
- The value $\alpha(11)=\frac{53}{10}$ confirms the conjecture and completes the known table through $d=16$.
- The formula genuinely stops at $m=d+1$: an explicit construction gives a smaller threshold just below the block, so the boundary is not an artifact of the method.
Reading between the lines
- A stability version is plausible: complexes whose edge-to-vertex ratio is close to the threshold should be near the explicit \u201cdeleted power set\u201d construction, with most edges concentrated on a small number of conglomerates.
- The weighted Katona lemma is a transferable tool: because it works for arbitrary monotone weights on order ideals, it may apply to other extremal problems on traces of set families.
- A natural next step is to map the values of $\alpha$ just above powers of two; the same mountain-and-conglomerate argument, with adjusted degree bounds, is the obvious tool to try.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the threshold function α(d), defined as the largest real constant such that every simplicial complex S with |S| ≤ α(d)|V(S)| has a vertex of degree at most d. The main theorem, Theorem 1.1, determines α(2^d − m) = (2^{d+1} − m)/(d + 1) for all integers d ≥ m ≥ 1, thereby resolving the entire block of arguments [2^d − d, 2^d]. The companion Theorem 1.2 proves α(11) = 53/10, confirming a conjecture of Frankl and Watanabe. Upper bounds are obtained from explicit simplicial complexes via Lemma 2.1; the lower-bound proof is organized around a more general statement for mountains, Theorem 5.1, and uses a weighted Kruskal–Katona lemma, local estimates in Lemmas 4.1–4.3, and a conglomerate decomposition. Section 6 adapts the same strategy to prove α(11).
Significance. If the proof is correct, the paper closes a natural block of previously open cases around powers of two and settles the α(11) conjecture. The upper-bound constructions are simple and transparent, and the lower-bound machinery is substantial and internally coherent: the dependency from Theorem 5.1 to Theorem 1.1 is clear, the small-case table matches known values, and the proof of Theorem 1.2 is a genuine extension beyond the m ≤ d regime. The proof is long and not machine-checked, so independent verification is advisable, but I found no load-bearing gap in the central chain. The main unsupported items are peripheral: the introduction asserts α(17) = 50/7 and α(20) = 8 without proof or reference, and the relation to the independent work of Li, Ma, and Rong is not made precise.
minor comments (4)
- [§1, after Theorem 1.2] The assertions α(17) = 50/7 and α(20) = 8 are presented as checked but are not proved or referenced; since these values do not follow from Theorems 1.1 or 1.2, they should be proved, referenced, or explicitly marked as conjectural.
- [§1, abstract and introduction] The statement that similar results were obtained independently by Li, Ma, and Rong [6] does not specify which cases overlap with Theorem 1.1 or Theorem 1.2; please state the precise overlap and, if applicable, the extent to which the proofs are independent.
- [§6, proof of Claim 6.2] The verification that the modified tuples in the two cases of Claim 6.2 satisfy the hypothesis of Theorem 6.1 is quite compressed, especially the degree bookkeeping for the vertex z in the second case; adding a few sentences explaining the degree changes would improve readability.
- [§1, table of values] The table of values for d ≤ 16 would benefit from a sentence explaining how each entry follows from Theorem 1.1, Theorem 1.2, and the previously known results cited in the introduction.
Circularity Check
No significant circularity: the proof derives matching upper and lower bounds from explicit constructions and a minimal-counterexample argument, not from fitted or assumed target values.
full rationale
I walked the claimed derivation chain. Theorem 1.1 is established by matching bounds: the upper bound comes from Construction 2 plus Lemma 2.1, which builds an explicit simplicial complex with the stated size-to-vertex ratio; the lower bound comes from Theorem 5.1, applied to the mountain of edges of size at least two. Theorem 5.1 is proved by a minimal-counterexample argument in which Claim 5.2 forces low-degree vertices, Claims 5.3, 5.7, 5.8, 5.12, and 5.13 assemble local estimates, and those local estimates are proved via the weighted Kruskal-Katona lemma (Lemma 3.8) and Lemmas 4.1-4.3. None of these steps assumes the target formula; they derive the coefficient 2^{d+1-m}/(d+1) from the assumed minimum-degree condition and from the sizes of the relevant shadow-initial segments. Theorem 1.2 is handled the same way: Construction 3 gives the upper bound 53/10, and Theorem 6.1 gives the lower bound through its own minimal-counterexample proof and Claims 6.2-6.4. The paper's prior work [7] is used as methodological inspiration and context, not as a load-bearing theorem; the previously known cases are re-proved in Examples 3.11-3.12 and Corollary 3.14 via Frankl's lemma and explicit constructions. The only questionable passages are the unproved assertions that α(17)=50/7 and α(20)=8 were 'checked', and the unquantified statement of similarity to Li-Ma-Rong [6]; these are peripheral and concern completeness or provenance, not circularity. I found no instance where a prediction is equivalent to its input by construction, no fitted parameter is renamed as a prediction, and no self-citation is used to force the central claim.
Assumptions & free parameters
assumptions (2)
- standard math Kruskal-Katona theorem (initial-segment shadow minimization)
- domain assumption Previously proved exact values alpha(2^d-1) and alpha(2^d-2)
Cite this review
Pith. "Pith review of Minimum degree in simplicial complexes." pith.science (2026). https://pith.science/paper/7DHD33T5
@misc{pith2026250101294,
author = {Pith},
title = {Pith review of: Minimum degree in simplicial complexes},
year = {2026},
howpublished = {\url{https://pith.science/paper/7DHD33T5}},
note = {Machine review of arXiv:2501.01294}
}
abstract
Given $d\in\mathbb{N}$, let $\alpha(d)$ be the largest real number such that every abstract simplicial complex $\mathcal{S}$ with $0<\vert\mathcal{S}\vert\leq\alpha(d)\vert V(\mathcal{S})\vert$ has a vertex of degree at most $d$. We extend previous results by Frankl, Frankl and Watanabe, and Piga and Sch\"ulke by proving that for all integers $d$ and $m$ with $d\geq m\geq 1$, we have $\alpha(2^d-m)=\frac{2^{d+1}-m}{d+1}$. Similar results were obtained independently by Li, Ma, and Rong.
Forward citations
Cited by 1 Pith paper
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Recent advances in arrow relations and traces of sets
A review of recent advances in arrow relations and traces of sets, presenting known theorems, constructions, and open problems without new results.
Reference graph
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Reviewed August 10, 2026 · model on record in the stance chip above.
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