{"id":"a52902f9-5c6d-4463-a377-5dec731ba35d","arxiv_id":"2606.22828","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"ex(n, C_{4k+2}^{4-}) equals the number of edges in the complete odd-bipartite 4-graph for large n, and the Turán density of C_{4k+2}^4 is 1/2 for all k≥2 with stability.","lead":"The paper proves that for large n the maximum edges in a 4-uniform hypergraph avoiding a tight even cycle of length 4k+2 minus one edge is achieved by the complete odd-bipartite construction. This extends earlier results on the expanded triangle and gives the density 1/2 plus stability for the full cycle when k is at least 2.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Dependence of 'sufficiently large n' on k left unquantified in stability/supersaturation steps","rationale":"The reader's weakest_assumption directly identifies the same quantitative gap; the full text does not appear to supply an explicit bound or growth rate, which is the single load-bearing point for the 'every k' statement.","tokens_in":1716,"tokens_out":334,"duration_ms":16221,"concrete_test":"Locate the stability theorem (likely the main result in §3 or §4) and extract the explicit function N(k) or the tower height arising from the removal lemma / supersaturation constants; recompute the implied threshold for k=2 and k=3 and verify whether it remains primitive recursive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts that for every fixed k ≥ 1 there exists N(k) such that the unique extremal example for ex(n, C_{4k+2}^{4-}) is the complete odd-bipartite 4-graph when n > N(k), and that this yields π(C_{4k+2}^4) = 1/2 for k ≥ 2. The argument relies on stability plus supersaturation to rule out other constructions. Because the forbidden subgraph has length linear in k, any application of the hypergraph removal lemma or iterative supersaturation inherits a dependence on k whose growth rate is not stated. If that dependence is super-exponential, the result remains formally true but the uniformity in k (the paper's main strengthening of Sankar) rests on an unverified quantitative assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that for every integer k ≥ 1 and all sufficiently large n, the unique extremal 4-uniform hypergraph on n vertices avoiding a tight 4-cycle of length 4k+2 minus one edge is the complete odd-bipartite 4-graph. As a consequence it obtains the Turán density π(C_{4k+2}^4) = 1/2 together with stability for every k ≥ 2, extending the Frankl–Keevash–Sudakov result on the expanded triangle and improving Sankar’s theorem (which required k sufficiently large).","tokens_in":1899,"tokens_out":546,"duration_ms":12874,"significance":"If correct, the result supplies an exact extremal construction and density for an infinite family of 4-uniform even cycles (minus an edge) that was previously known only asymptotically in k. The stability statement is a standard strengthening in the area and the reduction to the odd-bipartite construction is the central new contribution.","major_comments":[{"comment":"Introduction, paragraph following the statement of the main theorem: the claim that the result strengthens Sankar by removing the “sufficiently large k” hypothesis rests on the existence of N(k) for every fixed k ≥ 2. The stability and supersaturation arguments invoked to obtain this N(k) necessarily pass through the hypergraph removal lemma (or an iterative supersaturation step) whose quantitative bounds depend on the length 4k+2; no explicit bound or growth rate for N(k) is supplied, leaving open whether the improvement over Sankar is uniform in the stated sense.","section":"Introduction"},{"comment":"The proof that the complete odd-bipartite 4-graph is extremal for C_{4k+2}^{4-} (the minus-one-edge version) is used to deduce the density result for the full cycle C_{4k+2}^4. If the stability argument for the minus-one-edge problem contains a k-dependent error term that is not controlled uniformly, the deduction that π(C_{4k+2}^4) = 1/2 for every k ≥ 2 would require an additional limiting argument that is not indicated in the abstract.","section":"Main theorem statement"}],"minor_comments":[{"comment":"The abstract cites the Frankl–Keevash–Sudakov and Sankar results but does not give their precise statements or reference numbers; adding these in the introduction would clarify the exact strengthening.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the detailed comments. We address the two major comments point by point below.","responses":[{"response":"The main result is stated existentially: for every fixed integer k ≥ 1 there exists N = N(k) such that the unique extremal hypergraph on n > N vertices is the complete odd-bipartite 4-graph. This removes Sankar’s hypothesis that k itself must exceed some absolute constant, which is the strengthening claimed in the manuscript. The dependence of the quantitative bounds (via the removal lemma) on the cycle length 4k+2 is standard and does not invalidate the per-k existential statement. The manuscript makes no claim of a uniform N independent of k.","revision_made":"no","referee_comment":"[Introduction] Introduction, paragraph following the statement of the main theorem: the claim that the result strengthens Sankar by removing the “sufficiently large k” hypothesis rests on the existence of N(k) for every fixed k ≥ 2. The stability and supersaturation arguments invoked to obtain this N(k) necessarily pass through the hypergraph removal lemma (or an iterative supersaturation step) whose quantitative bounds depend on the length 4k+2; no explicit bound or growth rate for N(k) is supplied, leaving open whether the improvement over Sankar is uniform in the stated sense."},{"response":"For each fixed k ≥ 2 the stability theorem for the minus-one-edge problem supplies an error term that may depend on k. The Turán density π(C_{4k+2}^4) is the limit of ex(n, C_{4k+2}^4)/binom(n,4) as n → ∞ with k held fixed; any k-dependent error therefore disappears in this limit. The deduction therefore requires no additional limiting process in which k tends to infinity, and the abstract statement for all k ≥ 2 is justified by fixing k first.","revision_made":"no","referee_comment":"[Main theorem statement] The proof that the complete odd-bipartite 4-graph is extremal for C_{4k+2}^{4-} (the minus-one-edge version) is used to deduce the density result for the full cycle C_{4k+2}^4. If the stability argument for the minus-one-edge problem contains a k-dependent error term that is not controlled uniformly, the deduction that π(C_{4k+2}^4) = 1/2 for every k ≥ 2 would require an additional limiting argument that is not indicated in the abstract."}],"tokens_in":1465,"tokens_out":558,"duration_ms":33019,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper claims to resolve the extremal function and stability for the 4-uniform tight cycle of length 4k+2 minus one edge for every k ≥ 1. It also gets the Turán density 1/2 for the full cycle when k ≥ 2, with stability, for all such k.\n\nThe advance is that it covers every k instead of only large k as in Sankar, and it recovers the expanded triangle as the k=1 case. The proof strategy appears to adapt the stability methods from the triangle paper to the general cycle.\n\nWhat works is the clean statement that the odd-bipartite construction is extremal for the minus-one-edge version across the family. The stability result for all k is the part that was missing before.\n\nThe soft spot is the dependence of the \"sufficiently large n\" on k. The argument uses stability plus supersaturation, which typically produce thresholds that grow with the size of the forbidden subgraph. Since the cycle length grows with k, the threshold likely depends on k, but the paper does not quantify how. If the growth is reasonable the result is strong; if it is tower-like the uniformity is mostly formal.\n\nThis is for specialists in extremal set theory and hypergraph Turán problems. A reader who follows the Frankl-Keevash-Sudakov and Sankar papers will get value from seeing how the small k cases are handled. The work is grounded enough to deserve a serious referee, even if the quantitative aspects need checking in the proofs.","headline":"The paper fills in the small-k cases for the minus-one-edge Turán problem on 4-uniform even cycles and gets stability for all k.","tokens_in":2412,"tokens_out":388,"would_cite":true,"duration_ms":16988,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For large n the extremal 4-uniform hypergraphs avoiding a tight even cycle of length 4k+2 minus one edge are the complete odd-bipartite 4-graphs.","keywords":["Turán numbers","4-uniform hypergraphs","tight cycles","odd-bipartite hypergraphs","extremal graph theory","stability","hypergraph Turán problem"],"falsifier":"An explicit 4-uniform hypergraph on n vertices (n larger than the paper's implicit threshold for a given k) that contains more edges than any complete odd-bipartite 4-graph yet still contains no copy of C_{4k+2}^{4-}.","tokens_in":2631,"feed_emoji":"","tokens_out":775,"duration_ms":15149,"temperature":0.7,"pith_summary":"The paper proves that for every fixed k at least 1 and all sufficiently large n, the largest number of edges in a 4-uniform hypergraph without a copy of the tight cycle C_{4k+2}^4 minus one edge is attained precisely by the complete odd-bipartite 4-graphs. This construction is shown to be the unique extremal example and yields the Turán density 1/2 for the full cycle C_{4k+2}^4 when k is at least 2, together with a stability statement that any hypergraph with nearly that many edges must be close in structure to an odd-bipartite example. The result recovers the known density for the 4-uniform expanded triangle as the k=1 case and improves an earlier density theorem that held only for large k. A reader cares because the work fixes the asymptotic maximum edge density for an infinite family of forbidden subhypergraphs and supplies the structural information needed to understand near-extremal examples.","feed_headline":"Odd-bipartite 4-graphs maximize edges without cycle minus one edge","feed_subtitle":"They give the exact Turán number for C_{4k+2}^{4-} when n is large, forcing the density of the full cycle to 1/2 for every k ≥ 2.","key_machinery":"The complete odd-bipartite 4-graph, the unique maximum-edge construction shown to avoid every copy of C_{4k+2}^{4-} while attaining the stated density.","core_discovery":"For every integer k ≥ 1 and sufficiently large n, the extremal construction for the Turán number of the 4-uniform tight cycle of length 4k+2 minus one edge is a complete odd-bipartite 4-graph. In particular the Turán density of C_{4k+2}^4 is 1/2 for all k ≥ 2 together with the corresponding stability result.","pith_inferences":["The stability result may be strong enough to determine the exact Turán number for every n once the threshold is made explicit.","The same odd-bipartite construction could serve as the extremal example for other even-length tight cycles in higher uniformities.","Removing one edge from the cycle appears to be the minimal change that makes the density drop from its conjectured value to exactly 1/2."],"forward_implications":["The Turán density of every C_{4k+2}^4 equals 1/2 for k ≥ 2.","Any 4-uniform hypergraph with edge count within o(n^4) of the extremal number must be edit-close to a complete odd-bipartite 4-graph.","The same extremal construction works for the expanded-triangle case, recovering and extending the Frankl–Keevash–Sudakov theorem.","Stability transfers to the full cycle problem, so near-maximal hypergraphs without C_{4k+2}^4 are also close to odd-bipartite examples."],"fun_headline_variants":["Odd-bipartite 4-graphs extremal for C_{4k+2}^{4-} Turán numbers","Turán density of C_{4k+2}^4 equals 1/2 for k >= 2","Stability result for density of 4-uniform even cycles","Complete odd-bipartite 4-graphs give Turán number for C_{4k+2}^{4-}"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That n is large enough for stability and supersaturation arguments to rule out all other constructions.","fun_headline_variants_meta":{"raw":{"variants":["Odd-bipartite 4-graphs extremal for C_{4k+2}^{4-} Turán numbers","Turán density of C_{4k+2}^4 equals 1/2 for k >= 2","Stability result for density of 4-uniform even cycles","Complete odd-bipartite 4-graphs give Turán number for C_{4k+2}^{4-}"]},"model":"grok-4.3","cost_usd":0.007889,"raw_usage":{"total_tokens":3512,"prompt_tokens":659,"num_sources_used":0,"completion_tokens":94,"cost_in_usd_ticks":78890500,"prompt_tokens_details":{"text_tokens":659,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2759,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":659,"tokens_out":94,"duration_ms":16789,"temperature":1.0,"reasoning_tokens":2759,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T08:10:49.820399+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit 4-uniform hypergraph on n vertices (n larger than the paper's implicit threshold for a given k) that contains more edges than any complete odd-bipartite 4-graph yet still contains no copy of C_{4k+2}^{4-}.","supporting_citations":[],"review_version":1}