{"id":"b56b6d6b-1d32-4fdc-ab0f-070a0527b541","arxiv_id":"2606.20085","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A structural theorem for large cross-intersecting pairs is proved by extending Kupavskii's result, using a new S_{U,V}^Q-shift that preserves global and local intersection properties, yielding analogues of classical theorems.","lead":"The paper proves a structural theorem characterizing large cross-intersecting families of sets and introduces a new shifting technique that preserves intersection properties. A generalist might read it to see how stability ideas from intersecting families extend to pairs of families with applications in combinatorial design.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"S_{U,V}^Q-shift preservation of local substructures is the least-secured step in the structural characterization.","rationale":"The reader's weakest_assumption directly identifies the same technical step that the abstract flags as the key ingredient. Because the full text was not supplied for line-by-line inspection of the shift definition and its proof, the concern remains exactly where the reader placed it; no additional internal inconsistency is visible from the given material.","tokens_in":1711,"tokens_out":309,"duration_ms":9437,"concrete_test":"Take the smallest non-trivial parameters where diversity can appear (e.g., n=6, k=3) and construct an explicit cross-intersecting pair (A,B) whose diversity part is non-empty; apply the S_{U,V}^Q-shift as defined in the paper and verify whether every local intersection pattern required by the structural statement is retained after the shift.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim extends Kupavskii's structural theorem to cross-intersecting pairs by characterizing extremal examples via diversity parts and maximal extensions. This extension is said to rest on the new S_{U,V}^Q-shift, which is asserted to preserve both the global cross-intersecting property and certain local substructures. If the local-substructure preservation fails on any family whose diversity part is non-trivial, the inductive or shifting argument used to reach the claimed extremal forms cannot be completed, leaving the structural theorem unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript establishes a structural theorem for large cross-intersecting pairs of families on [n], extending Kupavskii's theorem from the intersecting to the cross-intersecting setting. It characterizes extremal pairs via their diversity parts and maximal cross-intersecting extensions. The proof relies on a new shifting operation, the S_{U,V}^Q-shift, which is claimed to preserve both the global cross-intersecting property and certain local substructures. Corollaries recover cross-intersecting analogues of the Han--Kohayakawa and Huang--Peng theorems.","tokens_in":1829,"tokens_out":400,"duration_ms":27015,"significance":"If the claimed preservation properties of the new shift hold, the result supplies a useful structural stability theorem in the cross-intersecting regime and introduces a shifting technique that the authors indicate has already been applied to a product version of the Hilton--Milner theorem. Such tools are valuable in extremal set theory for deriving further stability and product-type results.","major_comments":[{"comment":"The central argument rests on the assertion that the S_{U,V}^Q-shift preserves local substructures even when the diversity part is non-trivial (see the description of the shift in the proof of the main structural theorem). The provided sketch does not explicitly verify this preservation on families whose diversity part is non-empty; if the local-substructure claim fails in such cases, the inductive step used to reach the extremal forms cannot be completed.","section":"Proof of the structural theorem (shift preservation step)"}],"minor_comments":[{"comment":"The abstract states that the new shift 'maintains certain local substructures,' but the precise definition of the local substructures being preserved is not restated in the statement of the main theorem; adding a short explicit list would improve readability.","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 constructive comment on our manuscript. We address the major comment below.","responses":[{"response":"We agree that the sketch provided in the proof of the structural theorem does not contain an explicit verification of the local-substructure preservation property in the case of a non-empty diversity part. This omission leaves the inductive step insufficiently justified in the current version. In the revised manuscript we will expand the relevant section to include a complete case analysis: we will enumerate the possible configurations of the diversity sets relative to U, V and Q, and verify directly that the S_{U,V}^Q-shift preserves the required local intersection relations in each case. This addition will make the argument self-contained and complete the induction.","revision_made":"yes","referee_comment":"The central argument rests on the assertion that the S_{U,V}^Q-shift preserves local substructures even when the diversity part is non-trivial (see the description of the shift in the proof of the main structural theorem). The provided sketch does not explicitly verify this preservation on families whose diversity part is non-empty; if the local-substructure claim fails in such cases, the inductive step used to reach the extremal forms cannot be completed."}],"tokens_in":1333,"tokens_out":276,"duration_ms":23528,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's main contribution is a structural theorem that characterizes large cross-intersecting pairs by their diversity parts and maximal extensions, directly extending Kupavskii's earlier result on intersecting families. It also derives cross-intersecting versions of the Han-Kohayakawa and Huang-Peng theorems as corollaries.\n\nThe new S_{U,V}^Q shift is presented as the key technical tool. It is claimed to preserve both the global cross-intersecting property and certain local substructures, which allows the stability argument to go through. If that preservation holds, the method looks reusable, and the authors already note its use in a product Hilton-Milner result.\n\nThe potential weak point is exactly the local-substructure preservation step. The abstract asserts it works, but if the shift fails to maintain the required local properties on families with non-trivial diversity, the inductive or shifting argument for the full characterization would not close. That is the part that needs the most scrutiny in the proofs.\n\nThis is a paper for specialists in extremal set theory who already know the intersecting-family stability literature. A reader familiar with Hilton-Milner and Kupavskii will see the extension clearly and can judge whether the new shift adds a usable technique.\n\nThe work is coherent on its own terms and shows honest engagement with the prior results. It deserves a serious referee.","headline":"Extends Kupavskii's structural theorem to cross-intersecting pairs via a new shift operator, but the local-substructure preservation claim needs checking.","tokens_in":2320,"tokens_out":352,"would_cite":false,"duration_ms":16736,"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":"A structural theorem characterizes large cross-intersecting pairs by their diversity parts and maximal extensions.","keywords":["cross-intersecting families","extremal combinatorics","structural theorem","shifting method","diversity","stability"],"falsifier":"A pair of large cross-intersecting families whose structure cannot be described in terms of diversity parts and maximal cross-intersecting extensions would serve as a counterexample.","tokens_in":2610,"feed_emoji":"🔄","tokens_out":492,"duration_ms":7638,"temperature":0.7,"pith_summary":"The paper proves a structural result for the largest cross-intersecting pairs of families, extending an earlier theorem of Kupavskii on intersecting families. It describes these extremal pairs in terms of diversity parts and maximal cross-intersecting extensions. The argument rests on a new shifting operation that keeps both global intersection properties and local substructures intact. This yields cross-intersecting versions of several known stability theorems as direct corollaries.","feed_headline":"Structural theorem classifies large cross-intersecting pairs","feed_subtitle":"New shift preserves local substructures and yields analogues of classical stability results.","key_machinery":"The S_{U,V}^Q-shift, a new shifting method that preserves global intersection properties and certain local substructures after shifting.","core_discovery":"Large cross-intersecting pairs are characterized by their diversity parts and maximal cross-intersecting extensions. The characterization is obtained by applying the S_{U,V}^Q-shift, which preserves both global intersection properties and certain local substructures.","pith_inferences":["The shift operator may extend to other stability questions involving multiple families.","Similar structural results could appear for weighted or multipartite cross-intersecting settings."],"forward_implications":["Cross-intersecting analogues of the Han--Kohayakawa theorem hold.","Cross-intersecting analogues of the Huang--Peng theorem hold.","The same shifting technique applies to a product version of the Hilton--Milner theorem.","The structural description applies to all extremal pairs in the cross-intersecting setting."],"fun_headline_variants":["Large cross-intersecting pairs classified by diversity parts","New shift reveals structure of cross-intersecting pairs","Cross-intersecting families characterized by diversity and extensions","Structural theorem classifies cross-intersecting pair stability"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The new S_{U,V}^Q-shift preserves both global intersection properties and certain local substructures after shifting.","fun_headline_variants_meta":{"raw":{"variants":["Large cross-intersecting pairs classified by diversity parts","New shift reveals structure of cross-intersecting pairs","Cross-intersecting families characterized by diversity and extensions","Structural theorem classifies cross-intersecting pair stability"]},"model":"grok-4.3","cost_usd":0.00661,"raw_usage":{"total_tokens":3058,"prompt_tokens":613,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":66099500,"prompt_tokens_details":{"text_tokens":613,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2385,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":613,"tokens_out":60,"duration_ms":24332,"temperature":1.0,"reasoning_tokens":2385,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T17:02:56.659001+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A pair of large cross-intersecting families whose structure cannot be described in terms of diversity parts and maximal cross-intersecting extensions would serve as a counterexample.","supporting_citations":[],"review_version":1}