{"id":"a347f416-246b-400e-903a-c031e5597b11","arxiv_id":"2608.05568","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fixed d at least 4, the maximum independence number of a (d-1)-sphere on n vertices is n minus Theta(n^{1/floor(d/2)}); for flag 4- and 5-spheres it is n minus Theta(sqrt(n)), and Bier spheres have independence number at most half their vertices.","lead":"The paper proves sharp asymptotic bounds on the largest independent set in the graphs of high-dimensional simplicial spheres and flag spheres. It also shows that Bier spheres, a very large family of spheres, can never contain an independent set larger than half their vertices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Per-link g-vector identity in Lemma 3.3 is asserted without proof; the β upper bound hinges on its exact coefficients.","rationale":"The paper's main quantitative claim, β(d,n)=n−Θ(n^{1/⌊d/2⌋}), depends entirely on the upper bound in Lemma 3.3. The lower bound is a straightforward construction, but the upper bound hinges on (3.3) and (3.4), where a linear combination of face numbers of each vertex link is transformed into a weighted sum of g-vector entries. This transformation is asserted in one sentence and not derived. The reader's weakest assumption identified the dependency on g-nonnegativity, which is a standard theorem; my concern is different and more concrete: the algebraic identity itself is not shown, and a subtle coefficient error would break the argument. I verified the identity for d=4 and d=6 with simple examples, and it holds, but the general case is exactly the kind of alternating-sum manipulation where a small mistake is plausible. The proposed computational check would settle this definitively. I do not see other load-bearing issues: the flag-sphere arguments in Section 4 are detailed and check out, the Bier sphere proof is elementary and correct, and the lower-bound constructions are valid. The missing Appendix certificate only supports a peripheral counterexample, not the central theorems. Therefore, my recommendation is conditional acceptance: the identity in Lemma 3.3 should be either proved explicitly or verified symbolically before full acceptance.","tokens_in":12884,"tokens_out":50323,"duration_ms":374166,"concrete_test":"Use a computer algebra system (e.g., Sage or Sympy) to verify the per-link identity for arbitrary h-vectors of a (d-2)-sphere for d=4 through 12: substitute f_{i-2}(L)=Σ_j binom(d-1-j, i-1-j) h_j, impose Dehn-Sommerville h_j=h_{d-1-j}, and compare the left and right sides of (3.3) and (3.4). Alternatively, test on random simplicial spheres, such as boundary complexes of cyclic polytopes, by computing both sides for each vertex link. If every d matches, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.2's upper bound (Lemma 3.3) relies on an unstated identity, appearing in (3.3) and (3.4), that rewrites a face-number combination of each vertex link as (1/2)f_{d-2}(L) plus weighted g-vector terms. The text only remarks 'Here we used (3.1) and Lemma 2.3,' but does not derive the identity. A single wrong binomial coefficient or weight would invalidate the lower bound (d/2)|I| on the right-hand side, and the Kruskal–Katona contradiction would collapse. The reader verified this identity only for small cases; the general alternating sum is nontrivial and involves Dehn–Sommerville relations. Since this identity is load-bearing for the central β(d,n) claim, it deserves an explicit proof or a machine check.","agreement_with_reader":"partial"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nCitation 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.\n\nThis 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.","headline":"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.","tokens_in":13643,"tokens_out":2040,"would_cite":true,"duration_ms":19080,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E45","52B70"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["independence number","simplicial spheres","flag spheres","Bier spheres","g-vector","Dehn-Sommerville relations","stellar subdivision","Kruskal-Katona theorem"],"falsifier":"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.","tokens_in":12706,"feed_emoji":"🔺","tokens_out":15797,"duration_ms":169897,"temperature":0.7,"pith_summary":"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 $n\\ge3$.","feed_headline":"Independent sets in simplicial spheres reach n − Θ(n^{1/⌊d/2⌋})","feed_subtitle":"The answer is n minus a power of n; flag spheres match in dimensions 4 and 5; Bier spheres never beat n/2.","key_machinery":"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.","core_discovery":"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 $n\\ge3$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the original conjecture for flag spheres and the stellar-subdivision lower-bound construction for general spheres.","marker":"[6]"},{"why":"gives a prior flag-sphere construction with large independent sets, which this paper strengthens.","marker":"[17]"},{"why":"states the Upper Bound Theorem used to bound face numbers of arbitrary simplicial spheres.","marker":"[18]"},{"why":"gives the nonnegativity of the g-vector, the key hypothesis in the upper-bound argument for beta.","marker":"[2]"},{"why":"provides an additional proof of g-vector nonnegativity for simplicial spheres used in Lemma 2.3.","marker":"[12]"},{"why":"proves the flag upper bound theorem in dimension 3, motivating the join-of-two-cycles construction.","marker":"[20]"},{"why":"yields the edge and gamma-2 lower bounds for flag 3-spheres used in the d=4 and d=5 upper bounds.","marker":"[7]"},{"why":"shows edge stellar subdivision preserves flagness, validating the construction of Gamma_{d,s}.","marker":"[16]"},{"why":"defines Bier spheres and establishes they are combinatorial spheres, the setting of Theorem 1.4.","marker":"[5]"}],"fun_headline_variants":["Independence number of simplicial spheres is n minus a power of n","Flag spheres in dims 4,5: independence number is n−Θ(√n)","Bier spheres: independent sets never exceed half the vertices","Sharp bounds for independence numbers of simplicial spheres","Max independent set in simplicial spheres: n−Θ(n^{1/⌊d/2⌋})"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Independence number of simplicial spheres is n minus a power of n","Flag spheres in dims 4,5: independence number is n−Θ(√n)","Bier spheres: independent sets never exceed half the vertices","Sharp bounds for independence numbers of simplicial spheres","Max independent set in simplicial spheres: n−Θ(n^{1/⌊d/2⌋})"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001293,"raw_usage":{"total_tokens":5301,"prompt_tokens":987,"completion_tokens":4314,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":4209}},"tokens_in":603,"tokens_out":4314,"duration_ms":26950,"temperature":1.0,"reasoning_tokens":4209,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T10:24:54.069148+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Stable sets in flag spheres.European Journal of Combinatorics, 110:Paper No","cited_arxiv_id":null,"evidence_quote":"supplies the original conjecture for flag spheres and the stellar-subdivision lower-bound construction for general spheres."},{"cited_title":"Large Independent Sets in Flag Spheres","cited_arxiv_id":"2606.16109","evidence_quote":"gives a prior flag-sphere construction with large independent sets, which this paper strengthens."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"states the Upper Bound Theorem used to bound face numbers of arbitrary simplicial spheres."},{"cited_title":"On the anisotropy theorem of Papadakis and Petrotou.Algebraic Combinatorics, 6(5):1313–1330, 2023","cited_arxiv_id":null,"evidence_quote":"provides an additional proof of g-vector nonnegativity for simplicial spheres used in Lemma 2.3."},{"cited_title":"The flag upper bound theorem for 3- and 5-manifolds.Israel Journal of Mathematics, 222(1):401–419, 2017","cited_arxiv_id":null,"evidence_quote":"proves the flag upper bound theorem in dimension 3, motivating the join-of-two-cycles construction."},{"cited_title":"Davis and Boris Okun","cited_arxiv_id":null,"evidence_quote":"yields the edge and gamma-2 lower bounds for flag 3-spheres used in the d=4 and d=5 upper bounds."},{"cited_title":"Lutz and Eran Nevo","cited_arxiv_id":null,"evidence_quote":"shows edge stellar subdivision preserves flagness, validating the construction of Gamma_{d,s}."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines Bier spheres and establishes they are combinatorial spheres, the setting of Theorem 1.4."}],"review_version":1}