{"id":"dd00af90-d4d7-4219-b96e-531898ff40f6","arxiv_id":"1908.01744","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"An L-close Sperner family on an n-element set has at most sum_{h=0}^{|L|} C(n,h) members, and at most n members when L contains exactly one positive integer.","lead":"For collections of sets where every two sets differ by a small number of elements on the smaller side, this paper proves new upper bounds on how large the collection can be. It also gives simpler proofs of two known extremal results in this area of combinatorics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 1.2 is sound; the only dependency is the cited Lemma 19 in the secondary reproof.","rationale":"The reader's weakest-assumption analysis correctly identifies the positivity of L as necessary for p_{F,L}(v_F) != 0 and notes the cited Lemma 19 in the reproof of Theorem 1.4. I agree with both observations, but neither undermines the central claim. Theorem 1.2 is fully proved in the paper: the triangular evaluation condition in Lemma 1.3 is verified with the non-increasing ordering of set sizes, and the moreover part for |L|=1 uses a valid descending induction with no hidden sign error. The positivity of L is not an oversight; the paper explicitly separates the case where 0 is allowed and notes that the m <= n conclusion fails for L={0}. The unproved Lemma 19 is standard and affects only the secondary reproof of the Frankl-Furedi-Pach theorem, not the main extremal bound. Therefore the appropriate verdict is unchanged: ACCEPT with high confidence remains justified.","tokens_in":7292,"tokens_out":37595,"duration_ms":376474,"concrete_test":"Enumerate all 1-close Sperner families of three i-subsets of [n] for small n (say n <= 7 and 1 <= i <= n-1), i.e., triples with pairwise symmetric difference 2, and verify that each triple is either a star with common intersection of size i-1 or a triangle whose union has size i+1; this directly checks the unproved Lemma 19 of [5] used in the proof of Theorem 1.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the polynomial construction and the triangularization in Theorem 2.1, I find no load-bearing gap in Theorem 1.2. The ordering by decreasing |F_i| makes p_{F_i,L}(v_{F_j}) = prod_{h in L}(|F_i \\setminus F_j| - h) vanish for j < i, because then |F_i| <= |F_j| and the L-close property yields |F_i \\setminus F_j| = sd(F_i,F_j) in L; p_{F_i,L}(v_{F_i}) = prod_{h in L}(-h) != 0 requires 0 not in L, exactly as the paper assumes. The |L|=1 argument is also sound: descending induction shows all coefficients in a hypothetical dependence with the constant polynomial have one common value per size and are negative, forcing positive variable coefficients that cannot cancel; excluding the family {emptyset} is precisely to guarantee a variable appears. The only notable dependency is the unproved Lemma 19 of [5] used in the reproof of Theorem 1.4; it is a standard fact about cliques in the Johnson graph and does not bear on Theorem 1.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies L-close Sperner families in 2^[n], where every pair of distinct sets has skew distance sd(F,G)=min(|F\\G|,|G\\F|) belonging to a fixed set L of positive integers. Theorem 1.2 bounds such a family by sum_{h=0}^{|L|} C(n,h), and by n when |L|=1. The proof assigns to each set F a multilinear polynomial p_{F,L}, orders the sets by non-increasing size, and uses Lemma 1.3 to conclude linear independence; the |L|=1 case additionally shows that the polynomials remain independent together with the constant polynomial, via a descending induction on set sizes. The paper also gives a new proof of the Frankl-Furedi-Pach theorem exsd(n,{0,1}) = C(n,2)+2n-1 by induction, and closes with remarks on trace-Sperner families and a q-ary generalization.","tokens_in":7534,"tokens_out":17486,"duration_ms":172556,"significance":"If correct, Theorem 1.2 genuinely generalizes the Boros-Gurvich-Milanič result: it gives the linear bound m ≤ n for every single positive skew distance, and a Frankl-Wilson-type bound for arbitrary L. The polynomial proof is elegant and essentially self-contained: Lemma 1.3 is proved, the multilinearization step is valid, and the sign argument for |L|=1 is sound, with the {∅} edge case explicitly handled. The reproof of Theorem 1.4 is a nice additional contribution, though it imports Lemma 19 of [5] rather than proving it in the text. The sharpness examples (all k-subsets for L={1,...,k} and projective planes for L={q}) are appropriate and strengthen the paper.","major_comments":[],"minor_comments":[{"comment":"The existence of representative sets C_i for uniform levels with |F_i| ≥ 3 is delegated to 'an exercise for the reader (see Lemma 19 in [5])'; since the proof of Theorem 1.4 depends on this structural fact, the authors should either state and prove the needed Johnson-graph clique lemma or give a precise citation with statement.","section":"Proof of Theorem 1.4"},{"comment":"The identity p_{F',L}(v_F) = |F'|-|F| for |F'|>|F| is terse. The equality is correct because |F'|>|F| implies |F'\\F| > |F\\F'|, so the min equals |F\\F'| = s and hence |F'\\F| - s = |F'|-|F|; adding this one-line explanation would improve readability.","section":"Section 2, proof of Theorem 2.1, equation (2)"},{"comment":"In the line 'C(3,2)+2·3−1 = 2 3' the displayed value appears garbled: the expression equals 8, which is |2^[3]| and makes the base case trivial. Please correct the typographical rendering.","section":"Proof of Theorem 1.4, base case"},{"comment":"The definition of L-close Sperner is given twice, in the abstract and in the introduction; the later definition of L-sd for sets possibly containing 0 is clear, but a sentence in the introduction explicitly noting that 0 ∉ L is essential for Theorem 1.2 would help prevent the reader from applying the result to L={0}, for which the conclusion fails.","section":"Abstract and Introduction"}],"recommendation":"minor_revision","confidential_remarks":"The main theorem is new, correct, and proved cleanly; I have no concerns about the novelty or the citation pattern. The only weakness is that the reproof of the known Frankl-Furedi-Pach theorem depends on an unproved external lemma, but this is not load-bearing for the central claim and can be fixed by a short proof or a precise statement of the lemma."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zoli,\n\nShort version: this is a correct, useful little paper. The main new result is Theorem 1.2: for L-close Sperner families with L consisting of positive integers, the maximum size is at most sum_{h=0}^{|L|} C(n,h), and when |L|=1 it's at most n. The |L|=1 case generalizes Boros-Gurvich-Milanič's linear independence bound for 1-close Sperner systems to any single skew distance, and it's sharp via singletons and projective planes. The proof is the standard triangularization polynomial method: list sets by decreasing size, define p_{F,L} as the multilinearized product over h in L of (|F| - v_F·x - h), and observe the evaluation matrix is triangular with nonzero diagonal. That part is sound. The |L|=1 moreover part uses a Blokhuis-style argument to include the constant polynomial and rule out dependence; the descending induction on levels works, and the only excluded case is the family {∅}, which is indeed a genuine exception. I checked the algebra on examples; it holds up.\n\nThe paper also gives a new proof of the Frankl-Füredi-Pach theorem for {0,1}-skew-distance systems. The proof is inductive and reasonably clean, but it relies on a structural fact that it dismisses as 'an exercise' and cites as Lemma 19 of [5]. For a theorem that's already known, that's a minor self-containedness gap, not a correctness issue.\n\nSoft spots are minor. There are two index typos in the proof of Theorem 2.1: the sentence about vanishing should say 'if |F_i| ≤ |F_j|' and '|F_i \\ F_j| ∈ L' and conclude p_{F_i}(v_{F_j}) = 0. As printed, the inequality and subscripts are at least one of those things off, though the intended meaning is clear from context. Fix that before publication. Also, the paper doesn't address L containing 0 for the main theorem, but it explicitly explains why that's not a technicality (chains give n+1 for L={0}), so that's fair.\n\nNo data, no fitting, no circularity. The citations are appropriate, including their own work; the self-citations are for context, not to prop up the result. The Q_n conjecture is a nice hook for future work.\n\nWho is this for? People working in extremal set theory, especially the polynomial method. It will not change the field, but it's a solid, citable result with a clean proof. I'd send it to a good combinatorics journal, not desk reject. A serious referee should be able to verify it in an afternoon. Recommend: accept with minor corrections.","headline":"A correct, genuinely new generalization of the BGM 1-close bound to arbitrary singleton skew distances and a Frankl-Wilson type bound for general L; minor typos but sound.","tokens_in":8060,"tokens_out":11976,"would_cite":true,"duration_ms":101139,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D05","05A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that any family of subsets of an $n$-element set whose pairwise skew distances are confined to a fixed set $L$ of positive integers has at most as many members as the total number of subsets of size at most $|L|$, and…","keywords":["L-close Sperner","skew distance","antichain","multilinear polynomial method","linear independence","extremal set theory","trace Sperner families"],"falsifier":"Look for a counterexample with $L=\\{0\\}$: the maximal chain $\\emptyset\\subset\\{1\\}\\subset\\{1,2\\}\\subset\\dots\\subset[n]$ has $n+1$ sets and every distinct pair has skew distance $0$, so it violates the $|L|=1$ conclusion $m\\leq n$ if $0$ were admitted. This pinpoints the exact spot where the positivity of $L$ is doing the work, since the product $\\prod_{h\\in L}(-h)$ in the proof is the only obstruction.","tokens_in":7113,"feed_emoji":"🧮","tokens_out":9545,"duration_ms":88503,"temperature":0.7,"pith_summary":"The paper proves a size bound for families of subsets of $[n]$ whose pairwise skew distances all lie in a fixed set $L$ of positive integers: such a family has at most $\\sum_{h=0}^{|L|} \\binom{n}{h}$ members, and if $L$ is a single integer, at most $n$ members. This recovers and extends the earlier linear-independence bound for 1-close Sperner families and gives a skew-distance analogue of the classical $L$-intersection bound. The proof is a short polynomial method: each set is assigned a multilinear polynomial that vanishes on every other member when the family is ordered by decreasing size, so the polynomials must be linearly independent. The paper also settles the case $L=\\{0,1\\}$, where comparability is allowed, by proving the exact maximum is $\\binom{n}{2}+2n-1$.","feed_headline":"Skew-distance Sperner families capped by subset count","feed_subtitle":"A short multilinear-polynomial argument gives the bound and reproves the exact {0,1} case.","key_machinery":"The central object is the multilinear polynomial $p_{F,L}(x)=\\prod_{h\\in L}(|F|-v_F\\cdot x-h)$, obtained from the corresponding ordinary polynomial by replacing every $x_i^t$ with $x_i$. Its key evaluation identity is $p_{F,L}(v_G)=\\prod_{h\\in L}(|F\\setminus G|-h)$, which is nonzero when $F=G$ and zero for every pair with $|F\\setminus G|\\in L$. Ordering the family by non-increasing size makes the evaluation matrix triangular, so the independence lemma applies and the dimension of the space of multilinear polynomials of degree at most $|L|$ — namely $\\sum_{h=0}^{|L|}\\binom{n}{h}$ — bounds the family size. For $|L|=1$, the same evaluation identity is combined with a coefficient argument that shows the scaled polynomials cannot express the constant function $1$, giving the stronger $m\\leq n$.","core_discovery":"On its own terms, the paper's discovery is a triangular polynomial separation for skew-distance families: if $\\{F_1,\\dots,F_m\\}$ is $L$-close Sperner with $L\\subseteq [n]$ consisting of positive integers, then, listed in non-increasing order of size, the polynomials $p_{F_i,L}$ vanish on the characteristic vectors of earlier sets and not on their own, forcing linear independence. Counting the ambient space of multilinear polynomials of degree at most $|L|$ gives $m \\leq \\sum_{h=0}^{|L|} \\binom{n}{h}$. For a single allowed distance $L=\\{s\\}$, the same identity shows the polynomials together with the constant polynomial $1$ are independent, so $m\\leq n$. When $0$ is allowed, the paper reproves the exact result that a $\\{0,1\\}$-skew-distance family in $2^{[n]}$ has at most $\\binom{n}{2}+2n-1$ members, via an induction that splits the family into two pieces and controls one of them by a chain-like representative argument.","pith_inferences":["The polynomial separator is generic: applying the paper's set-encoding to the $q$-ary poset $Q_n$ produces $L$-close Sperner families in a larger cube, so a linear $O_q(n)$ upper bound for $\\{1\\}$-close families in $Q_n$ — a conjecture recorded in the paper — would follow from the same dimension argument rather than a bespoke construction.","The observation that $L=\\{\\ell+1,\\dots,n\\}$-close Sperner families are exactly $(n-\\ell)$-trace Sperner families transfers Theorem 1.2 into an explicit upper bound for trace problems, connecting skew-distance families to a classical invariant.","The $0$-in-$L$ case is structurally different: the exact $\\{0,1\\}$ value exceeds the positive-integer sum bound, so any full understanding of $ex_{sd}(n,L)$ for $0\\in L$ will need new machinery rather than a tweak of the polynomial count."],"forward_implications":["For $L=\\{1,2,\\dots,k\\}$ (the $k$-close Sperner property), the bound $m\\leq \\sum_{h=0}^k \\binom{n}{h}$ is asymptotically sharp as $n$ grows, witnessed by the uniform family of all $k$-subsets.","For a single positive skew distance $s$, no such family can have more than $n$ sets; the bound is tight for $s=1$ via all singletons, and for prime-power $s$ via the lines of a projective plane when $n=s^2+s+1$.","The exact maximum size of a $\\{0,1\\}$-skew-distance family in $2^{[n]}$ is $\\binom{n}{2}+2n-1$ for every $n\\geq 3$, matching the construction that layers a chain onto a near-uniform family.","Because the bound depends only on $|L|$, families with scattered allowed distances behave like families with a consecutive block of the same length, a direct analogue of the $L$-intersection phenomenon."],"supporting_citations":[{"why":"Supplies the original 1-close Sperner linear-independence theorem that Theorem 1.2 generalises, and the structural lemma used in the {0,1} reproof.","marker":"[5]"},{"why":"Establishes the classical L-intersection upper bound whose form Theorem 1.2 reproduces in the skew-distance setting.","marker":"[10]"},{"why":"Provides the extra idea used to prove the |L|=1 strengthening m ≤ n, namely treating the constant polynomial.","marker":"[3]"},{"why":"States the {0,1}-skew-distance extremal theorem that the paper reproves by induction.","marker":"[9]"},{"why":"Gives Sperner's theorem, the original antichain bound that motivates the definition of L-close Sperner systems.","marker":"[13]"},{"why":"Supplies the polynomial-method background that the triangular independence argument relies on.","marker":"[2]"}],"fun_headline_variants":["Single skew distance caps family size at n","Exact bound for skew distances 0 and 1","At most n sets if skew distance is fixed","Skew distances {0,1} give exact maximum size"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that every allowed skew distance is a positive integer: the proof needs $p_{F,L}(v_F)=\\prod_{h\\in L}(-h)$ to be nonzero, and when $0\\in L$ the conclusion $m\\leq n$ fails, as a chain of length $n+1$ shows.","fun_headline_variants_meta":{"raw":{"variants":["Single skew distance caps family size at n","Exact bound for skew distances 0 and 1","At most n sets if skew distance is fixed","Skew distances {0,1} give exact maximum size"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001034,"raw_usage":{"total_tokens":4346,"prompt_tokens":930,"completion_tokens":3416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":3353}},"tokens_in":546,"tokens_out":3416,"duration_ms":25702,"temperature":1.0,"reasoning_tokens":3353,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:06:06.401388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a counterexample with $L=\\{0\\}$: the maximal chain $\\emptyset\\subset\\{1\\}\\subset\\{1,2\\}\\subset\\dots\\subset[n]$ has $n+1$ sets and every distinct pair has skew distance $0$, so it violates the $|L|=1$ conclusion $m\\leq n$ if $0$ were admitted. This pinpoints the exact spot where the positivity of $L$ is doing the work, since the product $\\prod_{h\\in L}(-h)$ in the proof is the only obstruction.","supporting_citations":[{"cited_title":"Boros, V","cited_arxiv_id":null,"evidence_quote":"Supplies the original 1-close Sperner linear-independence theorem that Theorem 1.2 generalises, and the structural lemma used in the {0,1} reproof."},{"cited_title":"Frankl, R.M","cited_arxiv_id":null,"evidence_quote":"Establishes the classical L-intersection upper bound whose form Theorem 1.2 reproduces in the skew-distance setting."},{"cited_title":"Blokhuis, A new upper bound for the cardinality of 2-distance s ets in Euclidean space, Ann","cited_arxiv_id":null,"evidence_quote":"Provides the extra idea used to prove the |L|=1 strengthening m ≤ n, namely treating the constant polynomial."},{"cited_title":"Frankl, Z","cited_arxiv_id":null,"evidence_quote":"States the {0,1}-skew-distance extremal theorem that the paper reproves by induction."},{"cited_title":"Sperner, Ein Satz ¨ uber Untermengen einer endlichen Menge, Math","cited_arxiv_id":null,"evidence_quote":"Gives Sperner's theorem, the original antichain bound that motivates the definition of L-close Sperner systems."},{"cited_title":"Babai, P","cited_arxiv_id":null,"evidence_quote":"Supplies the polynomial-method background that the triangular independence argument relies on."}],"review_version":1}