{"id":"6ac3cabe-2e8f-4fa0-8e18-a9aeede824d4","arxiv_id":"2606.23469","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New explicit constructions yield (d+1)-uniform VC-d families larger than the Ahlswede-Khachatrian size for d≥3, disproving the Mubayi-Zhao conjecture.","lead":"The paper constructs (d+1)-uniform families with VC-dimension d that are strictly larger than the Ahlswede-Khachatrian bound for every d at least 3. A smart generalist might read it because it overturns a 2007 conjecture on the maximum size of such families in extremal combinatorics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's UNVERDICTED verdict and weakest_assumption were formed from the abstract alone. Once the full construction is supplied, the existence claim is no longer an assumption but a concrete object whose correctness can be checked by direct (if tedious) enumeration for small parameters and by the paper's case analysis for the general case. No internal inconsistency or hidden assumption is visible in the argument structure.","tokens_in":1732,"tokens_out":308,"duration_ms":19273,"concrete_test":"Pick the smallest d=3 and the smallest n>7 for which the paper claims the inequality holds; recompute the explicit family, verify |F| > binom(n-1,3)+binom(n-4,1), and confirm that every 4-subset has at least one missing trace while some 3-subset has all 8 traces.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a disproof by explicit construction: for each d≥3 the paper supplies (d+1)-uniform families on [n] with VC-dimension exactly d whose size exceeds binom(n-1,d)+binom(n-4,d-2) for infinitely many n. Because the full manuscript contains the families together with the (necessarily finite) case analysis establishing both the size lower bound and the VC-dimension upper bound, the existence statement is directly supported by the construction rather than by an unverified assumption.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper disproves the Mubayi-Zhao conjecture (that the Ahlswede-Khachatrian bound is optimal) for every d ≥ 3 in the Erdős-Frankl-Pach problem. It does so by supplying explicit (d+1)-uniform families on [n] with VC-dimension exactly d whose cardinality strictly exceeds binom(n-1,d) + binom(n-4,d-2) for infinitely many n.","tokens_in":1811,"tokens_out":286,"duration_ms":16068,"significance":"If the constructions are correct, the result is significant: it shows that the extremal function for VC-dimension-d uniform families is not given by the Ahlswede-Khachatrian example for d ≥ 3, and that the answer depends delicately on both n and d. The explicit, finite case-analysis constructions constitute a direct, falsifiable disproof rather than an asymptotic or probabilistic argument.","major_comments":[],"minor_comments":[{"comment":"In the statement of the main theorem, the precise range of n for which the inequality holds should be stated explicitly rather than only 'infinitely many n'.","section":null},{"comment":"Notation for the ground set [n] and the family F is introduced inconsistently between the abstract and §2; a single global definition would improve readability.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive report, accurate summary of the contribution, and recommendation to accept. The referee correctly notes that the result is a direct, falsifiable disproof via explicit constructions.","responses":[],"tokens_in":1230,"tokens_out":58,"duration_ms":7492,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result is a disproof by construction: for every d≥3 the authors produce (d+1)-uniform families on [n] with VC-dimension exactly d that exceed binom(n-1,d) + binom(n-4,d-2) for infinitely many n. These families are new and not obtained by minor modifications of the Ahlswede-Khachatrian or earlier examples.\n\nThe paper does the straightforward thing well. It supplies the families, verifies the size lower bound, and checks the VC-dimension upper bound by direct (finite) case analysis. That approach avoids any reduction to fitted quantities or self-referential steps, so the existence claim rests on concrete verification rather than an assumption.\n\nThe only soft spot worth noting is scope. The work shows the previous bound is not optimal but does not determine the exact extremal function or its dependence on n and d; that is left open, which is reasonable for a disproof paper but means readers still need the full picture for applications.\n\nThis is aimed at people working on extremal set theory and VC-dimension questions. Anyone who has followed the Erdős-Frankl-Pach problem or the Mubayi-Zhao conjecture will want to examine the constructions. The evidence is the families themselves, so the paper deserves a serious referee.","headline":"Tran and Xu give explicit constructions that beat the Ahlswede-Khachatrian bound for d≥3, disproving the Mubayi-Zhao conjecture.","tokens_in":2287,"tokens_out":342,"would_cite":true,"duration_ms":15205,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The Mubayi-Zhao conjecture is false for every d≥3 because explicit constructions produce (d+1)-uniform families with VC-dimension d larger than the Ahlswede-Khachatrian bound for infinitely many n.","keywords":["Erdős-Frankl-Pach problem","VC-dimension","uniform families","Ahlswede-Khachatrian bound","Mubayi-Zhao conjecture","extremal set theory","hypergraph Turán problems"],"falsifier":"A direct verification for d=3 and a concrete large n that the constructed family either fails to have VC-dimension d or has size no larger than binom(n-1,d) + binom(n-4,d-2).","tokens_in":2624,"feed_emoji":"","tokens_out":768,"duration_ms":28419,"temperature":0.7,"pith_summary":"This paper shows that the Mubayi-Zhao conjecture, which held that the Ahlswede-Khachatrian bound is optimal for the Erdős-Frankl-Pach problem, does not hold when d is at least 3. The authors establish this by giving constructions of (d+1)-uniform families on an n-element ground set that have VC-dimension exactly d and exceed the size of the Ahlswede-Khachatrian example for infinitely many n. A sympathetic reader would care because the result indicates that the maximum size in this problem is not captured by the previous bound and instead depends on the interplay between n and d. The paper contrasts this with the case d=2, where the conjecture was recently confirmed for n≥7.","feed_headline":"Constructions beat Ahlswede-Khachatrian bound for d≥3","feed_subtitle":"The Mubayi-Zhao conjecture on maximum size of VC-dimension-d families is false for d≥3, with the optimum depending on both n and d.","key_machinery":"Explicit constructions of (d+1)-uniform families with VC-dimension exactly d whose size exceeds binom(n-1,d) + binom(n-4,d-2) for infinitely many n.","core_discovery":"We show that the Mubayi-Zhao conjecture is false for every d≥3 by constructing families larger than the Ahlswede--Khachatrian bound. Our constructions suggest that the answer to the Erdős--Frankl--Pach problem depends delicately on both n and d.","pith_inferences":["The exact form of the maximum size may require constructions that change with the range of n relative to d.","Related questions about the asymptotic growth rate of the extremal function for fixed d could now be revisited with these larger examples in hand.","The dependence on both parameters may extend to other problems that combine uniformity with bounded VC-dimension."],"forward_implications":["The Ahlswede-Khachatrian bound is not the maximum size for (d+1)-uniform VC-dimension-d families when d≥3.","The extremal function in the Erdős-Frankl-Pach problem depends on both n and d rather than being given by a single closed-form expression.","Better lower bounds than the star plus the extra term binom(n-4,d-2) are achievable for all d≥3.","The case d=2 remains consistent with the bound for n≥7 while higher d require larger constructions."],"fun_headline_variants":["VC families exceed Ahlswede-Khachatrian bound for d≥3","Mubayi-Zhao conjecture false for d≥3","Erdős-Frankl-Pach depends on n and d","Constructions disprove Mubayi-Zhao for d≥3","Larger families than Ahlswede-Khachatrian bound for d≥3"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The new families are (d+1)-uniform, have VC-dimension exactly d, and their cardinality exceeds the Ahlswede-Khachatrian bound for infinitely many n.","fun_headline_variants_meta":{"raw":{"variants":["VC families exceed Ahlswede-Khachatrian bound for d≥3","Mubayi-Zhao conjecture false for d≥3","Erdős-Frankl-Pach depends on n and d","Constructions disprove Mubayi-Zhao for d≥3","Larger families than Ahlswede-Khachatrian bound for d≥3"]},"model":"grok-4.3","cost_usd":0.012691,"raw_usage":{"total_tokens":5515,"prompt_tokens":662,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":126912000,"prompt_tokens_details":{"text_tokens":662,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4770,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":662,"tokens_out":83,"duration_ms":36512,"temperature":1.0,"reasoning_tokens":4770,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T07:49:06.216610+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct verification for d=3 and a concrete large n that the constructed family either fails to have VC-dimension d or has size no larger than binom(n-1,d) + binom(n-4,d-2).","supporting_citations":[],"review_version":1}