{"id":"56123eef-5cd2-4131-a652-a70b4c30c2c2","arxiv_id":"2505.00130","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For n large, every n-vertex r-uniform hypergraph with r at least about n/2 and minimum degree above the sharp Dirac threshold contains Berge cycles of every length from 2 to n.","lead":"This paper proves exact minimum-degree conditions under which a large uniform hypergraph contains Berge cycles of every possible length, from 2 up to its number of vertices. It completes a line of Dirac-type theorems showing that the same degree threshold that forces one Hamiltonian cycle actually forces cycles of all lengths.","discovery_kind":"extension","skeptic_critique":null,"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves sharp minimum degree conditions for Berge pancyclicity in n-vertex r-uniform hypergraphs when r ≥ floor((n-1)/2)-1 and n is sufficiently large (n ≥ 31 for the lower-uniformity case, n ≥ 55 for the higher-uniformity case). The main theorem splits into two parts: one for r = floor((n-1)/2)-1, handled by a graph-compatibility argument (Theorem 10), and one for r ≥ floor((n-1)/2), handled via the incidence bipartite graph and two structural theorems (Theorems 9 and 8). The proof uses the Kostochka–Luo–McCourt Hamiltonian Berge cycle theorem as a black box, then studies the set of extra edges outside that cycle. Sharpness is witnessed by three explicit constructions whose minimum degrees fall one below the thresholds.","tokens_in":17429,"tokens_out":57877,"duration_ms":504760,"significance":"If the proof is completed, the result is significant: it provides exact Dirac-type thresholds for Berge pancyclicity in the large-uniformity regime, completing the earlier result of Bailey–Li–Luo and giving a hypergraph analogue of Bondy's meta-conjecture. The paper is well structured, and several ingredients are elegant and useful: the shifting-function lemma, the self-shift complementary set structure, and the cycle-compatibility framework. The sharpness constructions are natural and clearly explained. The reliance on an external published theorem is methodologically acceptable, though it means the new result inherits the correctness of that theorem.","major_comments":[{"comment":"The assertion that a k-SSC set contains an interval of d consecutive vertices is not justified and is false in general. For n=24, k=21, d=gcd(24,21)=3, the set A={v0,v2,v4,...,v22} satisfies A∩(k+A)=∅ and |A|=12, hence is k-SSC with 0∈A, but it contains no three consecutive vertices. The subsequent construction of the (k+1)-cycle requires two disjoint intervals of length d inside U'_0, so the proof needs an argument using the specific size-2 case m+p=d+1, not an appeal to the definition of k-SSC.","section":"Section 3, Case 4 (Theorem 9), after Claim 13"},{"comment":"The reduction \"Without loss of generality, assume {v0,...,v_{k-1}} = U'_0\" is not valid. For k=n/2, a k-SSC set is any transversal of the pairs {i, i+k}; for example with n=8, k=4, {v0,v1,v2,v7} is k-SSC but not a contiguous block. The subsequent edge-counting and Berge-cycle construction rely on contiguity, so this normalization must be proved or the argument must be adapted to arbitrary transversals.","section":"Section 3, Case 4, final subcase k=d=n/2"},{"comment":"The step concluding that the index set of U'_0 is k-SSC is missing an argument. The absence of a (k+1)-Berge cycle only rules out k-chords, i.e., pairs at distance k contained in a single extra edge; it does not by itself prevent U'_0 from containing two vertices at distance k in different edges. One must first use |U'_0|=r+1 and |E'_0|≥3 to show that every pair of vertices in U'_0 is contained together in some edge of E'_0, since each edge in E'_0 misses exactly one vertex of U'_0. This is easy to supply but is load-bearing for the SSC conclusion.","section":"Section 3, Case 4, paragraph after Claim 12"}],"minor_comments":[{"comment":"The displayed hamiltonian cycle in the proof of Lemma 11 is written as \"C = v0e0v1...vn−1en−1v1\"; the final vertex should be v0.","section":"Section 2.1"},{"comment":"The sentence \"By Theorem 8, if δ(G) ≥ m/3+4, then G is weakly pancyclic with girth 4\" should say \"weakly bipancyclic\" because G is bipartite; the intended conclusion for H is unaffected since only even cycles in G are relevant.","section":"Section 2, proof of Theorem 5"},{"comment":"The notation e_t is reused for an extra edge after having denoted a cycle edge of C; this is confusing and should be renamed, for instance e*.","section":"Section 4, Claims 16 and 17"},{"comment":"The sentence \"one can check that the minimum degrees of the graphs are one less than those in Theorem 5\" should say \"hypergraphs\" instead of \"graphs\".","section":"Section 1, Construction 1"},{"comment":"The inequality r(n/2−15)/n ≥ c_r for n ≥ 55 is asserted tersely; spelling out the two cases r=floor((n−1)/2) and r≥n/2 would improve readability.","section":"Section 2, proof of Theorem 5"}],"recommendation":"major_revision","confidential_remarks":"The two gaps in Case 4 of Theorem 9 are substantial and affect the load-bearing argument for the main pancyclicity claim. The first gap is a false general assertion about k-SSC sets; the second is an invalid WLOG reduction to a contiguous block. I believe both are fixable with additional work, but they require genuine revisions rather than copy-editing. The external black-box theorem (Kostochka–Luo–McCourt) is acceptable, and the paper's overall approach is sound outside Case 4. If the authors can repair Case 4, the paper would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for the report. I read the paper and think the reader's ACCEPT is right, with the same moderate confidence.\n\nThe new content is real: Theorem 5 completes the hypergraph pancyclicity story for r >= ~n/2 by proving exact thresholds in the previously open range. Theorems 9 and 10 are the engine: they show that a Hamiltonian Berge cycle plus enough extra edges forces all cycle lengths. That is a new structural mechanism, not just a tweak of the Hamiltonian proof.\n\nI went through the reductions and they hold. The incidence-graph argument in Section 2 is elegant and correct. The shifting-function lemma is clean. Theorem 10's cycle-compatibility setup (Claim 15) is a nice idea, and the counting in Claim 14 checks out. Theorem 9 is longer, but the cases are organized and the use of SSC sets is legitimate.\n\nSoft spots: The proof leans on Theorem 3 from Kostochka–Luo–McCourt without re-proving it. That's fine in practice—it's a published theorem—but it means the paper's validity is tied to that external result. The hardest part is Theorem 9 Case 4 (n=2r+2). Claims 12 and 13 are plausible, and the final interval/SSC arguments are intricate. I didn't find a hole, but this is exactly where I'd want a second pair of eyes. The paper's own n≥31 and n≥55 thresholds are not optimal, and the authors admit this; that doesn't affect the theorem. One small blemish: in the concluding remarks, they cite [11] as unavailable, so that part of the discussion is unverified.\n\nBottom line: This is a solid contribution, worth a serious referee. It should be published in a good combinatorics journal after a careful check of Case 4. I'll cite it if I write anything on Berge cycles.","headline":"A genuine completion of the large-r pancyclicity threshold, with sound but intricate proofs; referee should focus on Theorem 9 Case 4.","tokens_in":17985,"tokens_out":5273,"would_cite":true,"duration_ms":51853,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C35","05C38"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sharp minimum-degree conditions force Berge cycles of every length in large-uniformity hypergraphs.","keywords":["hypergraph","Berge cycles","pancyclic","minimum degree","extremal combinatorics","Hamiltonian cycles","r-uniform hypergraphs","bipancyclic"],"falsifier":"Search the boundary cases, for instance n = 31 with r = 14, for a 14-uniform hypergraph with minimum degree at least 106 that is missing a Berge cycle of some specific length; finding one would refute Theorem 5, while a proof that every such hypergraph has all lengths, and that the only degree-105 examples fail, would confirm the claimed sharpness.","tokens_in":17290,"feed_emoji":"🌀","tokens_out":8055,"duration_ms":73490,"temperature":0.7,"pith_summary":"The paper proves exact minimum-degree thresholds that force an n-vertex r-uniform hypergraph to contain Berge cycles of every length from 2 to n, for the large-uniformity range r ≥ floor((n-1)/2)-1 and sufficiently large n. The thresholds are the same ones that already force a Hamiltonian Berge cycle, so the result is a hypergraph analogue of the classical graph meta-conjecture that Hamiltonian strength should imply pancyclicity. Below r = n/2 the required minimum degree is binomial(floor((n-1)/2), r-1)+1; at or above r = n/2 it is simply r. Three extremal constructions show that lowering the minimum degree by one can destroy the Hamiltonian cycle, so the conditions are sharp. Together with earlier work for smaller r, this closes the pancyclicity question for all uniformities when n is large.","feed_headline":"Hypergraphs hit every Berge cycle length under one sharp degree bound","feed_subtitle":"Minimum-degree conditions that already force a Hamiltonian Berge cycle also force cycles of every length from 2 to n.","key_machinery":"The shifting function S_s maps vertex indices forward by s with a single fold, and Lemma 11 shows that if S_s(U_0) meets the Hamiltonian edge e_0, then H contains an (n-s+1)-Berge cycle; contrapositively, a missing cycle forces the disjointness condition S_s(U_0) ∩ e_0 = ∅. The k-self-shift-complementary (k-SSC) sets — subsets A of the n-cycle with |A| = n/2 and A ∩ (k+A) = ∅ — classify the vertex footprints that a large extra edge can have when no k-chord exists, and the SSC-structure proposition forces such an A to alternate inside cosets of a subgroup. In the borderline-uniformity case, a cycle-compatible graph is constructed whose edges correspond to pairs with large co-degree in the extra edges, and known weak-pancyclicity and weak-bipancyclicity results for graphs are applied to force cycles of every length in that graph, which then lift to Berge cycles.","core_discovery":"The central claim, Theorem 5, is that for r ≥ floor((n-1)/2)-1, every n-vertex r-uniform hypergraph meeting the stated minimum-degree condition contains Berge cycles of all lengths 2 through n. The proof begins with the Hamiltonian Berge cycle C guaranteed by the same degree condition, then studies the edges outside C. The key observation is that if such an extra edge contains a k-chord, the hypergraph immediately has a (k+1)-Berge cycle. The bulk of the argument shows that if some length were missing, the footprints of the extra edges would have to form a rigid alternating pattern, a k-self-shift-complementary set, whose structure contradicts the degree condition. For the borderline case r = floor((n-1)/2)-1, a cycle-compatible auxiliary graph is built and shown to be weakly pancyclic, so cycles in the graph lift to Berge cycles of the same length. The three boundary constructions fail to be Hamiltonian and have minimum degree exactly one below the thresholds, establishing sharpness.","pith_inferences":["The numerical thresholds n ≥ 31 and n ≥ 55 are probably not optimal; the proof leaves slack in the inequalities, so the first genuinely hard examples may occur at smaller n than the theorem assumes.","The paper's Question 2 — whether every sufficiently large Hamiltonian r-uniform hypergraph is pancyclic — is answered positively for r > n/2 and negatively when r divides n and n ≥ r^2+3r; the unresolved interval near r = sqrt(n(1-o(1))) is a natural next target for the cycle-compatibility method.","If the six-extra-edge condition in Theorem 9 could be lowered to one, the r = floor((n-1)/2) case would reduce to the authors' Question 1, which would give a very tight bond between Hamiltonicity and pancyclicity."],"forward_implications":["The large-uniformity range r ≥ floor((n-1)/2)-1 now has sharp minimum-degree conditions for pancyclicity, completing the picture together with the earlier small-uniformity result.","All three boundary constructions have minimum degree exactly one below the theorem's thresholds and fail even to be Hamiltonian, so no uniform weakening of the degree condition can force pancyclicity.","For r ≥ n/2, a single vertex lying in c_r extra edges beyond the Hamiltonian cycle suffices, with c_r = 1 when r > n/2 and c_r = 6 when r = floor((n-1)/2).","In the borderline case r = floor((n-1)/2)-1, the numerical condition n ≥ 31 can be relaxed to a structural one: each vertex in at least 5(r-1)+2 extra edges forces pancyclicity."],"supporting_citations":[{"why":"Supplies the Hamiltonian Berge cycle whose existence is used throughout; every constructed shorter cycle starts from this cycle and the extra edges.","marker":"[18]"},{"why":"Covers the complementary range r ≤ floor((n-1)/2)-2, so Theorem 5 completes the pancyclicity picture across all uniformities.","marker":"[2]"},{"why":"The classical graph result that a Hamiltonian graph with many edges is pancyclic, used in the cycle-compatibility argument.","marker":"[4]"},{"why":"Provides the weak bipancyclicity theorem for bipartite graphs applied to the incidence graph.","marker":"[16]"},{"why":"Provides the weak pancyclicity theorem for graphs used on the auxiliary graph in the borderline case.","marker":"[5]"}],"fun_headline_variants":["Berge cycles of every length from one degree condition","Sharp degree bound forces all Berge cycle lengths","Hypergraphs: one minimum degree triggers all Berge cycles","All Berge cycles from 2 to n under a sharp bound","Exact Dirac-type condition yields pancyclic hypergraphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof uses, without re-proving it, an external theorem saying that the same minimum-degree conditions already force a Hamiltonian Berge cycle; if that theorem had a flaw or different thresholds, Theorem 5 would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Berge cycles of every length from one degree condition","Sharp degree bound forces all Berge cycle lengths","Hypergraphs: one minimum degree triggers all Berge cycles","All Berge cycles from 2 to n under a sharp bound","Exact Dirac-type condition yields pancyclic hypergraphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1503,"prompt_tokens":907,"completion_tokens":596,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":517}},"tokens_in":523,"tokens_out":596,"duration_ms":5980,"temperature":1.0,"reasoning_tokens":517,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:53:33.101227+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search the boundary cases, for instance n = 31 with r = 14, for a 14-uniform hypergraph with minimum degree at least 106 that is missing a Berge cycle of some specific length; finding one would refute Theorem 5, while a proof that every such hypergraph has all lengths, and that the only degree-105 examples fail, would confirm the claimed sharpness.","supporting_citations":[{"cited_title":"Dirac-type theorem s for long Berge cycles in hypergraphs","cited_arxiv_id":null,"evidence_quote":"Supplies the Hamiltonian Berge cycle whose existence is used throughout; every constructed shorter cycle starts from this cycle and the extra edges."},{"cited_title":"Bailey, Y","cited_arxiv_id":null,"evidence_quote":"Covers the complementary range r ≤ floor((n-1)/2)-2, so Theorem 5 completes the pancyclicity picture across all uniformities."},{"cited_title":"Pancyclic graphs I","cited_arxiv_id":null,"evidence_quote":"The classical graph result that a Hamiltonian graph with many edges is pancyclic, used in the cycle-compatibility argument."},{"cited_title":"Weakly bipancyclic bipartite graphs","cited_arxiv_id":null,"evidence_quote":"Provides the weak bipancyclicity theorem for bipartite graphs applied to the incidence graph."},{"cited_title":"Sufficient conditions for graphs to contain all subgr aphs of a given type","cited_arxiv_id":null,"evidence_quote":"Provides the weak pancyclicity theorem for graphs used on the auxiliary graph in the borderline case."}],"review_version":1}