{"id":"ab4c3803-1f89-4c39-91dc-1ce532f4bd36","arxiv_id":"1908.01108","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the diamond poset D2, sat^*(n,D2) ≥ sqrt(n); for the (k+1)-element antichain, sat^*(n,A_{k+1}) ≥ (1-o_k(1)) k n / log_2 k.","lead":"Researchers improved the known minimum sizes of two types of forbidden-pattern families in the n-dimensional Boolean lattice. The new proofs give a square-root lower bound for the diamond pattern and a nearly (k n) / log_2 k lower bound for antichain patterns, beating earlier results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Wide-gap swap in Section 3 can create a larger wide gap in a third chain; Theorem 11(1) termination proof is incomplete.","rationale":"The diamond bound in Theorem 6 is well supported: Lemma 9's arc-counting is sound, and the distinctness of the n arcs follows from the unique removed element. The reader's color-class invariant concern in Part (2) is also answerable: if a gap element remained uncolored at the start of color class j, maximality of the greedy chain would force it to be added, contradicting that (X,Y) is a final gap; so it must already have an earlier color. The real soft spot is the termination of the wide-gap reduction in Part (1). The proof only controls gaps containing the entire triple X,T,Y. A gap in a third chain that contains T and X but not Y can become wide after the move and can have size larger than the previous maximum, so the asserted monotonicity of the maximum wide-gap size is not established. This is a serious proof gap because the final 'no wide gaps' partition is needed for the d ≤ floor(log_2 k)+1 bound used to derive Theorem 7. The abstract's overstatement about improving 3n−1 for all k ≥ 3 is a separate, minor defect. Since the proof gap is potentially repairable and no counterexample to the theorem itself is established, conditional acceptance is appropriate, but the condition should be a complete termination proof or an independent verification of that step, not merely a corrected comparison in the introduction.","tokens_in":7872,"tokens_out":35409,"duration_ms":350833,"concrete_test":"Settle the termination claim by searching for a concrete counterexample in B_n with n ≤ 6 and k around 20. Enumerate families F closed under gap-filling rule (ii) for a k-chain decomposition, with width at most k; run the prescribed swap on a maximum-size wide gap; check whether any wide gap of size strictly larger than the previous maximum appears. The target configuration is: C_i contains W ⊂ X ⊂ Y; C_j contains S ⊂ T ⊂ U inside (X,Y); and some third chain has a gap (A,B) with A ⊂ W ⊂ X ⊂ T ⊂ B, Y not ⊂ B, and |B-A| > |Y-X|. If such an instance exists, the swap can increase the maximum wide gap and the stated termination argument is false. If none exists for n ≤ 6, provide a computer-assisted proof of the missing monotonicity case or a repaired argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section 3, Part (1), the chain-reassignment argument claims that moving T from C_j to C_i for a maximum-size wide gap (X,Y) does not increase the maximum size among wide gaps. The proof explicitly discusses only gaps containing X, T, and Y. A gap (A,B) in a third chain C_l can contain T and X while omitting Y. Before the move, C_i ∩ (A,B) may contain two elements W and X, and C_j ∩ (A,B) may contain S and T, so the gap is not wide. After T moves to C_i, the same gap contains W, X, T and becomes wide. Its size |B-A| can exceed d = |Y-X| if A lies sufficiently far below X while B lies inside (X,Y) or just outside it. Item (ii) does not rule this out; it only forces all elements of the interval into F, and this is compatible with width k when k is large enough. The sentence 'any gap in some Ci′ containing X, T, Y must also contain S, T, U' does not cover gaps containing T and X but not Y. Since Theorem 11(1) is the bound used to obtain Theorem 7 for large k, the central antichain lower bound is not fully proved as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the minimum size sat*(n,P) of an induced-P-saturated family in the n-dimensional Boolean lattice. Theorem 6 establishes sat*(n,D2) ≥ ceil(sqrt n), improving the logarithmic lower bound of Ferrara et al.; the proof uses Lemma 9 to force, for each member F* and each coordinate i, a distinct arc in a digraph on the family, giving n arcs and hence |F| ≥ sqrt n. Theorem 7 establishes sat*(n,A_{k+1}) ≥ (1 − 1/log_2 k) k n/log_2 k for k ≥ 3 and n sufficiently large, via a Dilworth chain decomposition and two counting arguments: Theorem 11(1) repeatedly moves an element from a 'wide' gap between chains and then counts gap elements, while Theorem 11(2) builds greedy color classes and applies Sperner's theorem to gaps in each color class. The diamond proof is clean and self-contained; the antichain proof, however, has two steps that are not justified as written.","tokens_in":8065,"tokens_out":20378,"duration_ms":214425,"significance":"If the antichain proofs can be completed, the paper gives a substantial advance in induced poset saturation. The sqrt(n) lower bound for the diamond is the first polynomial lower bound for that poset and follows from a short, elegant arc-counting argument. The antichain lower bound is asymptotically k n/log_2 k, far above the previous 3n−1 bound and within a logarithmic factor of the known upper bound. The paper is written in an elementary style and makes explicit use only of classical results such as Dilworth's theorem and Sperner's theorem. The main weakness is not novelty but proof completeness in Section 3.","major_comments":[{"comment":"The argument that moving T from C_j to C_i does not increase the maximum size among wide gaps is incomplete: it analyzes only gaps in C_i and C_j, and only gaps that contain X, T, and Y. A gap (A,B) in a third chain C_l can contain X and T but not Y. For example, with A ⊂ W ⊂ X ⊂ T ⊂ B ⊂ Y, where W,X ∈ C_i and S,T ∈ C_j with U not in (A,B), the gap (A,B) has two elements from C_i and two from C_j before the move, hence is not wide; after T moves to C_i, it has three elements from C_i and becomes wide, and its size |B−A| can exceed |Y−X|. The sentence 'any gap in some C_i′ containing X,T,Y must also contain S,T,U' does not cover such a gap. Since the termination of the reassignment process depends on the maximum wide-gap size not increasing, Theorem 11(1) and consequently Theorem 7 are not fully proved as written.","section":"Section 3, Part (1), the wide-gap reassignment argument"},{"comment":"The assertion that for a gap (X,Y) in the chain forming color class j, every element of the open interval (X,Y) has already received one of the colors 1,...,j−1 is not justified. Item (ii) applies to gaps in the chains of a k-chain partition of F, but the greedy color class j is not shown to be a chain in such a partition; maximality of the greedy construction only excludes uncolored members of F from the gap, not elements of B_n outside F. The later Sperner bound, (d choose floor(d/2)) ≤ j−1, requires every element of the interval to be colored, so the proof of Theorem 11(2) is missing a step. This does not affect Theorem 7 if Part (1) is repaired, but it is a genuine gap in a stated result.","section":"Section 3, Part (2), the greedy color-class invariant"}],"minor_comments":[{"comment":"The definition says an arc (A,B) is included when |B−A|=1, but the proof writes arcs (F_i,G_i) where F_i−G_i={i}; the ordered-pair convention should be clarified, for instance by writing arcs from the smaller set to the larger set consistently.","section":"Section 2, proof of Theorem 6"},{"comment":"The phrase 'n ≥ log_2^3 k' should be typeset as n ≥ (log_2 k)^3 to avoid ambiguity.","section":"Section 3, final paragraph"},{"comment":"In Lemma 9, the proof of part (2) is dismissed with 'the proof for (2) follows similarly'; since the dual argument is not literally identical, a few sentences of detail would improve readability.","section":"Section 2, Lemma 9"}],"recommendation":"major_revision","confidential_remarks":"The diamond result is publishable as is. The main obstruction is the incomplete termination proof for the wide-gap reassignment in Part (1), which is load-bearing for the paper's headline antichain bound. The color-class invariant in Part (2) also needs a proof. Both issues appear local and potentially repairable, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nTwo things to know. First, the diamond result (Theorem 6) is clean and convincing: the arc-counting argument with Lemma 9 is a genuinely new device and the proof checks out. So we now have sat^*(n,D_2) ≥ √n, a real step past the logarithmic bound. Second, the antichain result (Theorem 7) is not fully proved as written. The chain-reassignment step in part (1) of Theorem 11 has a hole that looks load-bearing.\n\nThe issue is in the termination argument for eliminating wide gaps. They take a maximum-size wide gap (X,Y) in C_i, with S,T,U in C_j inside it, and move T to C_i. They claim the maximum size of a wide gap does not increase. Their proof only considers gaps containing X, T, and Y. But a gap (A,B) in a third chain C_l can contain T and X without containing Y. Before the move, C_i might contribute only two elements to that interval (say W and X) and C_j might contribute S and T, so the gap is not wide. After T moves, C_i has W, X, T and the gap becomes wide, and its size |B-A| can exceed d = |Y-X| if A lies enough below X and B lies just above T. The sentence about gaps containing X,T,Y doesn't cover this case. So the claim that the maximum wide gap size doesn't increase is unsupported, and with it the bound in part (1), which is the part that gives the advertised k n/log k for large k.\n\nThe abstract also overstates the antichain comparison: it says the new bound improves 3n-1 for k≥3, but for small k the coefficient is below 3. The paper's Theorem 7 quantifies 'n sufficiently large,' but the abstract doesn't say 'for sufficiently large k,' which is misleading.\n\nWhat's good: the diamond proof, as said, is solid. Part (2) of Theorem 11 might survive and gives a lower bound around k n / (log k)^{3/2}, which is still a real improvement over 3n−1 for large k, though weaker than claimed. The paper is clearly written, builds appropriately on Ferrara et al., and the citation pattern is fine.\n\nBottom line: this deserves serious peer review, but the antichain theorem needs a repaired proof or a weakened claim. The reader's take gave soundness 8/10 and didn't catch the wide-gap issue; I think the stress-test is correct. If you're working in this area, the diamond result is worth citing; the antichain result is a promising but incomplete direction.\n\nRecommendation: send to peer review with a request for major revision on the antichain part.","headline":"Solid diamond lower bound, but the antichain proof has a real, load-bearing gap in the wide-gap elimination argument that the paper does not address.","tokens_in":8641,"tokens_out":12523,"would_cite":true,"duration_ms":103400,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D05","06A07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that induced-saturated families in the Boolean lattice must be large: at least $\\lceil\\sqrt{n}\\rceil$ for the diamond poset and, for large $k$, about $k n/\\log_2 k$ for a $(k+1)$-element antichain.","keywords":["induced poset saturation","Boolean lattice","saturation number","diamond poset","antichain","chain decomposition","gap bounds","lower bounds"],"falsifier":"A single induced-$\\mathcal{D}_2$-saturated family $\\mathcal{F}\\subseteq 2^{[n]}$ with $|\\mathcal{F}|<\\lceil\\sqrt{n}\\rceil$ for some $n$ would refute the diamond theorem; a single induced-$\\mathcal{A}_{k+1}$-saturated family with fewer than $(1-1/\\log_2 k)\\, k n/\\log_2 k$ members, for $k\\geq 3$ and $n\\geq (\\log_2 k)^3$, would refute the antichain theorem. Inspecting the greedy coloring of such a family would also directly test the claimed invariant that uncolored elements cannot appear in a gap of a later color class.","tokens_in":7637,"feed_emoji":"💎","tokens_out":13163,"duration_ms":115495,"temperature":0.7,"pith_summary":"The paper establishes two lower bounds on induced poset saturation in the $n$-dimensional Boolean lattice, the minimum size of a family of subsets that contains no induced copy of a fixed poset but that would create one if any missing subset were added. For the four-point diamond $\\mathcal{D}_2$, the minimum size is at least $\\lceil\\sqrt{n}\\rceil$, improving a logarithmic lower bound. For the antichain with $k+1$ elements, the minimum size is at least $(1 - 1/\\log_2 k)\\, k n/\\log_2 k$ for each fixed $k\\geq 3$ and sufficiently large $n$, which for large $k$ improves the earlier linear bound of $3n-1$. These results show that even minimal 'trigger' families for two basic posets must contain many sets, and they narrow the gap to the known upper bounds of $n+1$ and roughly $kn$.","feed_headline":"At least √n sets are needed to saturate the diamond poset","feed_subtitle":"Antichain saturation grows to about k n / log k sets, nearly matching the known upper bounds.","key_machinery":"The main objects are the Boolean lattice $\\mathcal{B}_n=(2^{[n]},\\subseteq)$, the diamond $\\mathcal{D}_2$ (the four-element poset with two incomparable middle elements), and the antichain $\\mathcal{A}_{k+1}$. The diamond proof is carried by a directed cover graph on the family, with an arc $A\\to B$ when $B=A\\cup\\{i\\}$; the key lemma forces every coordinate $i$ to appear in at least one arc, so the graph has at least $n$ arcs although it has at most $|\\mathcal{F}|(|\\mathcal{F}|-1)$ arcs. The antichain proof is carried by a chain decomposition of the family, by the open intervals, called gaps, between consecutive elements of a chain, and by a greedy coloring that assigns each element to one of $k$ color classes; the size of any gap in color class $j$ is bounded by the largest $d$ with $\\binom{d}{\\lfloor d/2\\rfloor}\\leq j-1$, and this bound is what converts chain lengths into the linear lower bound.","core_discovery":"The paper's central results are Theorem 6 and Theorem 7. Theorem 6 states that every induced-$\\mathcal{D}_2$-saturated family in $\\mathcal{B}_n$ has size at least $\\lceil\\sqrt{n}\\rceil$; the proof builds a directed graph on the family whose arcs record pairs differing in exactly one coordinate, shows via a coordinate-by-coordinate lemma that each of the $n$ coordinates forces a distinct arc, and counts $|\\mathcal{F}|(|\\mathcal{F}|-1)\\geq n$. Theorem 7 states that for $k\\geq 3$ and $n$ sufficiently large, every induced-$\\mathcal{A}_{k+1}$-saturated family has size at least $(1 - 1/\\log_2 k)\\, k n/\\log_2 k$; the proof partitions the family into $k$ chains, uses the classical bound on antichains inside a Boolean lattice to control the length of gaps between consecutive chain elements, and sums the resulting lower bounds on chain lengths.","pith_inferences":["Nothing in the diamond proof uses the full force of saturation beyond the coordinate lemma; the same arc-counting idea may apply to other small posets, such as the butterfly, and would yield polynomial lower bounds if every coordinate can again be assigned a distinct cover pair.","If the unproved greedy-coloring invariant for the antichain can be established, the same template likely extends to induced saturation in products of chains, where the relevant antichain bound is known but takes a different form.","The paper itself closes by conjecturing the true values are $n+1$ for the diamond and $kn-o(k)$ for the antichain; if that is right, both lower bounds here are still far from optimal, and stronger arguments must exploit structure beyond single-coordinate counting and gap sizes."],"forward_implications":["Every induced-$\\mathcal{D}_2$-saturated family in $\\mathcal{B}_n$ has at least $\\lceil\\sqrt{n}\\rceil$ members, so no family of size $o(\\sqrt{n})$ can saturate the diamond.","For fixed $k\\geq 3$ and large $n$, $\\operatorname{sat}^*(n,\\mathcal{A}_{k+1})\\geq (1 - 1/\\log_2 k)\\, k n/\\log_2 k$; for $k$ large enough this is stronger than the previous $3n-1$ lower bound.","The antichain argument forces both $\\emptyset$ and $[n]$ to belong to any induced-$\\mathcal{A}_{k+1}$-saturated family, and every gap in a chain containing no chain elements must itself be entirely filled with family members.","The two parts of the antichain theorem are complementary: for fixed $k\\leq 243$ the gap-sum bound is stronger, while for $k\\geq 264$ the wide-gap bound wins, so the proof covers two different regimes."],"supporting_citations":[{"why":"Introduces induced poset saturation in the Boolean lattice and supplies the prior logarithmic diamond bound and the $3n-1$ antichain bound that the paper strengthens.","marker":"[4]"},{"why":"Supplies the chain-decomposition theorem used to partition the antichain-saturated family into $k$ chains.","marker":"[1]"},{"why":"Supplies the numerical bound on the largest antichain in a Boolean lattice that controls the allowed size of gaps.","marker":"[8]"}],"fun_headline_variants":["√n lower bound for diamond saturation","Diamond saturation needs at least √n sets","Antichain saturation leaps to kn/log k","Induced poset saturation bounds sharpened","From log n to √n: diamond saturation bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The antichain proof assumes that after the greedy coloring, every subset lying strictly between two consecutive members of color class $j$ has already received one of the earlier colors; this invariant is stated as a consequence of the coloring's structure without a full derivation.","fun_headline_variants_meta":{"raw":{"variants":["√n lower bound for diamond saturation","Diamond saturation needs at least √n sets","Antichain saturation leaps to kn/log k","Induced poset saturation bounds sharpened","From log n to √n: diamond saturation bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00094,"raw_usage":{"total_tokens":4022,"prompt_tokens":956,"completion_tokens":3066,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":2997}},"tokens_in":572,"tokens_out":3066,"duration_ms":21135,"temperature":1.0,"reasoning_tokens":2997,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:25:25.404295+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A single induced-$\\mathcal{D}_2$-saturated family $\\mathcal{F}\\subseteq 2^{[n]}$ with $|\\mathcal{F}|<\\lceil\\sqrt{n}\\rceil$ for some $n$ would refute the diamond theorem; a single induced-$\\mathcal{A}_{k+1}$-saturated family with fewer than $(1-1/\\log_2 k)\\, k n/\\log_2 k$ members, for $k\\geq 3$ and $n\\geq (\\log_2 k)^3$, would refute the antichain theorem. Inspecting the greedy coloring of such a family would also directly test the claimed invariant that uncolored elements cannot appear in a gap of a later color class.","supporting_citations":[{"cited_title":"Ferrara, B","cited_arxiv_id":null,"evidence_quote":"Introduces induced poset saturation in the Boolean lattice and supplies the prior logarithmic diamond bound and the $3n-1$ antichain bound that the paper strengthens."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the chain-decomposition theorem used to partition the antichain-saturated family into $k$ chains."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the numerical bound on the largest antichain in a Boolean lattice that controls the allowed size of gaps."}],"review_version":1}