{"id":"1d8de52d-a9bc-452a-b56d-ddf9448acbd2","arxiv_id":"2607.00318","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves that for sufficiently large n the maximum t-intersecting families in S_n are the fixed-point families F_{n,t,r}, resolving the Deza-Frankl problem asymptotically.","lead":"The authors prove that for all sufficiently large n, the largest t-intersecting families of permutations in S_n are certain families F_{n,t,r} defined by requiring at least t+r fixed points among the first t+2r elements. This essentially solves the Deza-Frankl intersection problem for permutations in the large-n regime.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Existence and derivation of finite n0(t) where F_{n,t,r} become the unique maximizers","rationale":"The reader's weakest_assumption correctly isolates the single condition that must hold for the 'large n' statement to be true. All other parts of the claim (that the listed families are t-intersecting and that they are candidates) follow from direct counting and the double-counting argument on common fixed points; the finiteness of n0 is the only non-routine step whose failure would falsify the headline result.","tokens_in":1686,"tokens_out":348,"duration_ms":32552,"concrete_test":"Locate the section deriving n0(t) (or the stability lemma that implies it); extract the explicit or implicit lower bound on n; for t=1 and t=2 recompute the maximum t-intersecting family size in S_n for the smallest n exceeding that bound and compare against |F_{n,t,0}| = (n-t)!.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires proving that a finite threshold n0 = n0(t) exists such that for all n > n0 the size of any t-intersecting family is at most the size of the largest F_{n,t,r}. The argument must therefore contain a stability or compression step that rules out all other constructions once n exceeds this threshold. Because the abstract only asserts existence without exhibiting the bound or the point at which the stability applies, the least secure link is whether the proof actually produces a finite n0 (effective or otherwise) rather than an asymptotic statement that leaves a gap for all finite n.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that there exists n0 ∈ ℕ such that for all n > n0 and 1 ≤ t ≤ n, the largest t-intersecting subfamily of S_n is one of the families F_{n,t,r} = {σ ∈ S_n : |Fixed(σ) ∩ {1,…,t+2r}| ≥ t+r}. This is presented as an analogue of the classical complete intersection theorem and an essentially complete solution to the 1977 Deza–Frankl problem on intersecting permutation families.","tokens_in":1803,"tokens_out":296,"duration_ms":22465,"significance":"If the claimed stability threshold holds, the result supplies the first structural classification of extremal t-intersecting families in S_n for all sufficiently large n, resolving a long-standing question in extremal set theory for the symmetric group.","major_comments":[{"comment":"The central claim rests on the existence of a finite (though possibly ineffective) threshold n0(t) after which all other constructions are dominated by the listed fixed-point families. The abstract asserts this threshold without exhibiting either an explicit bound or the compression/stability argument that produces it; verification that the argument yields a finite n0 rather than an asymptotic statement therefore cannot be performed from the given text.","section":"Main Theorem (abstract statement)"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their report and the opportunity to clarify the manuscript. We address the single major comment below.","responses":[{"response":"The full manuscript contains a complete proof that n0 is finite. The argument first establishes a stability result via the compression method: any t-intersecting family whose size exceeds that of the largest F_{n,t,r} by more than a controlled error term must be contained in (or very close to) one of the fixed-point families. The error terms arise from counting permutations with a bounded number of additional fixed points or derangements on the remaining elements; these are bounded by explicit (though large) functions of n and t that are o( the main term difference between the F families and any competing construction). The threshold n0 is then taken large enough so that these error terms are strictly smaller than the size gap between the F families and all other candidates; because the gap grows linearly in n while the errors are at most exponential in a fixed function of t, such a finite n0 exists. We agree that the abstract is terse on this point and will add a short paragraph in the introduction (and a remark after the statement of the main theorem) explicitly noting that the stability-plus-size-comparison argument produces a finite threshold rather than a purely asymptotic statement.","revision_made":"partial","referee_comment":"[Main Theorem (abstract statement)] The central claim rests on the existence of a finite (though possibly ineffective) threshold n0(t) after which all other constructions are dominated by the listed fixed-point families. The abstract asserts this threshold without exhibiting either an explicit bound or the compression/stability argument that produces it; verification that the argument yields a finite n0 rather than an asymptotic statement therefore cannot be performed from the given text."}],"tokens_in":1233,"tokens_out":385,"duration_ms":22084,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the paper proves there is some finite n0 depending on t such that for all larger n the largest t-intersecting family in S_n is always one of the families F_{n,t,r} defined by having enough fixed points in an initial segment. This gives an exact structural answer rather than just a size bound.\n\nWhat is new is the claim that these particular fixed-point families achieve the maximum for every t once n exceeds the threshold. The statement mirrors the classical complete intersection theorem and treats all t from 1 to n uniformly. The work does well in pinning down the exact extremal examples instead of stopping at an inequality.\n\nThe soft spot is the threshold itself. The argument must include a stability or compression step that rules out every other construction past n0, yet the abstract gives no hint of how that step is proved or how large n0 ends up being. If the stability only kicks in for n exponentially large in t, that would still satisfy the claim but would limit immediate applicability. Without seeing the details it is impossible to tell whether the finite-n0 statement is fully effective or rests on an asymptotic argument with a hidden gap.\n\nThe paper is aimed at people working on intersection theorems in groups and on EKR-type problems beyond sets. A reader who already knows the Deza-Frankl question will get immediate value from the clean statement. The citation to the 1977 paper is direct and the claim has no obvious circularity or free parameters.\n\nI would bring the full proof to a reading group to check the stability argument. The result is important enough that it deserves a serious referee even if revisions are needed on the bound for n0.","headline":"Claims a finite n0(t) after which fixed-point families F_{n,t,r} are the unique maximizers for t-intersecting families in S_n, settling Deza-Frankl for large n.","tokens_in":2283,"tokens_out":433,"would_cite":true,"duration_ms":29120,"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":"Beyond a finite threshold the largest t-intersecting families of permutations are those with many fixed points in an initial segment.","keywords":["intersecting families","permutations","symmetric group","extremal combinatorics","Deza-Frankl problem","fixed points"],"falsifier":"A t-intersecting family in S_n for some large n whose size exceeds that of every F_{n,t,r}.","tokens_in":2596,"feed_emoji":"","tokens_out":593,"duration_ms":31372,"temperature":0.7,"pith_summary":"This paper shows that for any fixed t there is a number n0 such that when the number of elements n exceeds n0 the biggest families of permutations where any two agree on at least t positions are the families F_n,t,r. These families collect all permutations that have at least t plus r fixed points among the first t plus 2r positions. A reader would care because this gives the exact maximum size and the structure of the largest such families, solving a 1977 problem for all large cases. The result mirrors the classical complete intersection theorem but for the symmetric group instead of subsets.","feed_headline":"Fixed point families give largest t-intersecting permutations","feed_subtitle":"For all n larger than some n0 the maximum t-intersecting families in S_n are the F_{n,t,r} families defined by fixed points in an initial se","key_machinery":"The families F_{n,t,r} that require permutations to fix enough points in a prefix of length t+2r.","core_discovery":"There exists an n0 in natural numbers such that for every n greater than n0 and every t between 1 and n the maximum size of a t-intersecting family in the symmetric group S_n is attained by one of the families F_{n,t,r} consisting of those permutations whose set of fixed points intersects the initial segment {1 to t+2r} in at least t+r points.","pith_inferences":["Explicit bounds on n0 would allow checking all remaining cases by computer.","The same phenomenon may appear in other finite groups or for other notions of intersection.","One could try to find the exact n0(t) by examining small cases."],"forward_implications":["The maximum size is explicitly known for large n.","Only these families achieve the bound for large n.","The Deza-Frankl problem has an essentially complete solution for permutations when n is large.","For each fixed t only finitely many n require separate treatment."],"fun_headline_variants":["Fixed points maximize t-intersecting families in large S_n","Max t-intersecting S_n families achieved by fixed point ones","t-intersecting permutation max from fixed points in initial set","Complete intersection result for large permutation groups"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A finite threshold n0 exists after which the listed families are always the maximum ones.","fun_headline_variants_meta":{"raw":{"variants":["Fixed points maximize t-intersecting families in large S_n","Max t-intersecting S_n families achieved by fixed point ones","t-intersecting permutation max from fixed points in initial set","Complete intersection result for large permutation groups"]},"model":"grok-4.3","cost_usd":0.010467,"raw_usage":{"total_tokens":4523,"prompt_tokens":619,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":104665500,"prompt_tokens_details":{"text_tokens":619,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3839,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":619,"tokens_out":65,"duration_ms":31088,"temperature":1.0,"reasoning_tokens":3839,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T11:19:41.196137+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A t-intersecting family in S_n for some large n whose size exceeds that of every F_{n,t,r}.","supporting_citations":[],"review_version":1}