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Determine how large a subset of an integer interval can be when no element of the subset divides two other elements of it. Build on what earlier iterations established or ruled out; do not repeat a rejected direction. Push the best upper and lower bounds you can justify, and formalise in Lean any lemma you actually prove.
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LITERATURE-01 read The paper investigates families of chains in a partially ordered set and establishes Sperner-type properties, i.e., combinatorial bounds on the size of families that avoid certain chain configurations. It develops results using classic extremal set theory tools such as the LYM inequality, Dilworth's theorem, and related combinatorial arguments.
LITERATURE-01 read The cited chapter 'Summatory Functions' is a general treatment of analytic number‑theoretic techniques for evaluating sums of arithmetic functions. It does not discuss extremal subsets of integer intervals with divisibility constraints, nor does it provide bounds or constructions for sets where no element divides two others.
LITERATURE-02 read The cited work is a graduate thesis entitled "Asymptotic Formulae for Restricted Unimodal Sequences". Only bibliographic metadata and no abstract or content are available, so no concrete results, methods, or relevance to divisibility‑constrained extremal subsets can be extracted.
LITERATURE-02 read The source is a figure (Figure 2) from a PeerJ article showing barnacle density and maximum body size. It contains no mathematical content related to integer intervals, divisibility constraints, or extremal set theory.
LITERATURE-02 read The paper presents an adaptive mesh‑free method for lower‑bound limit analysis formulated as a nonlinear programming problem. It focuses on computational mechanics and does not discuss combinatorial properties of integer intervals or divisibility constraints.
LITERATURE-01 read The chapter discusses decomposition theory for lattices that lack chain conditions, focusing on results such as Dilworth's theorem and methods for partitioning partially ordered sets into chains and antichains. It provides general lattice-theoretic techniques but does not address the specific problem of bounding subsets of an integer interval under a divisibility‑based restriction.
LITERATURE-02 read The thesis presents implementations of auctions using Lagrangian relaxation, interior‑point linear programming, and upper‑bound linear programming. It focuses on computational optimization methods for auction problems and does not discuss combinatorial number theory or divisor-free subsets of integer intervals.
LITERATURE-01 read The preprint proposes using Dynamic Mode Decomposition (DMD) to predict nuclide number densities in lattice physics calculations. It presents a data‑driven modeling approach for nuclear engineering applications and reports experimental validation on benchmark problems.
LITERATURE-02 read The chapter presents a bound on the number of weighted blow-ups required to compute the minimal log discrepancy for smooth threefolds, using techniques from birational geometry and the theory of weighted blow-ups.
LITERATURE-01 read The chapter surveys Turán-type extremal problems, presenting general methods (e.g., Turán's theorem, Erdős–Stone, hypergraph extensions, probabilistic constructions) for bounding the size of families that avoid a prescribed substructure. It discusses how to translate combinatorial forbidden configurations into graph or hypergraph settings and derive upper bounds, but provides no specific results on integer intervals or divisibility constraints.
LITERATURE-01 read The chapter introduces a combinatorial "counting sieve" method for estimating the size of families of integers that avoid prescribed divisibility configurations. It presents a general inclusion‑exclusion‑type framework that can be used to derive upper bounds for sets where certain divisor relations are forbidden.
LITERATURE-01 read The preprint presents a suite of algorithms for detecting isomorphism between partially ordered sets using a hierarchical matrix decomposition framework (Hierarchical Poset Matrix Tree). It details recursive decomposition, poset matrix duality, and canonical sub‑orderings, achieving empirical time complexities between O(n^2) and O(n^4). The work focuses on algorithmic performance and preservation of order‑theoretic invariants, without addressing extremal combinatorial questions.
LITERATURE-02 read The technical report by V. N. Temlyakov (2001) discusses lower bound estimates for greedy approximation algorithms. It presents analytical techniques for deriving two distinct lower estimates in the context of approximation theory, but it does not address integer intervals, divisibility constraints, or extremal set problems.
LITERATURE-01 read The paper introduces the divisor‑product graph MD(n) whose vertices are the proper divisors of a non‑prime integer n and where two vertices are adjacent when their product divides n. It studies connectivity, computes vertex degrees, and determines chromatic and clique numbers for n = p^α (α≥3), showing χ(MD(n)) = ω(MD(n)).
LITERATURE-02 read The paper investigates how many integers in a given set possess a large prime factor, using analytic and sieve‑theoretic methods. While it addresses divisibility properties of integers, it does not directly treat subsets where no element divides two others, nor does it give explicit extremal bounds for that condition.
LITERATURE-01 read The paper computes the degree distance of the zero‑divisor graph Γ[Z_n] for n = p^2, n = pq, and n = p^3 (p, q distinct primes). It focuses on graph‑theoretic invariants of zero‑divisor graphs of the ring Z_n and does not discuss subsets of integer intervals nor the combinatorial problem of bounding a set where no element divides two others.
LITERATURE-01 read The chapter "Algebraic methods in Sperner theory" surveys algebraic techniques used in extremal set theory, such as the Lubell–Yamamoto–Meshalkin inequality, eigenvalue arguments, and generating‑function methods, to bound the size of families that avoid certain inclusion relations. While the focus is on Sperner families (no set contains another), the discussed methods are generally applicable to other partially ordered sets.
LITERATURE-01 read The paper derives extensions of the Kraft inequality for lossy compression and uses them to obtain refinements of the Shannon lower bound in various rate‑distortion coding settings. It focuses on information‑theoretic inequalities and coding theorems, not on combinatorial properties of integer sets or divisor relations.
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Map each integer n≤N to its vector of prime exponents (e1,…,ek) truncated by log N. The condition ‘no element divides two others’ means no vector is component‑wise ≤ two distinct vectors. Thus the selected vectors form a 2‑wise antichain in a product of chains. Apply high‑dimensional Sperner and Lubell–Yamamoto–Meshalkin (LYM) inequalities to bound the size of such families, possibly sharpening via the Kruskal‑Katona theorem for product posets. This yields a clean analytic upper bound that improves on earlier spectral attempts.
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A multiplicative Sidon set forbids a·b = c·d with distinct pairs. Our property is weaker, but any Sidon set also satisfies it. Recent work gives tight asymptotics for maximal Sidon sets in [1,N] (≈N^{1/2}). We will examine the proof techniques (Erdős‑Turán, Fourier analysis on the multiplicative group) and relax the equality constraint to the divisor condition, aiming to construct larger sets (perhaps of size N^{2/3}) while preserving the needed property. This direction provides a constructive lower bound that complements the antichain upper bound.
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Empirically optimal sets for small N appear to consist of blocks of consecutive integers interspersed with gaps dictated by the smallest divisor present. We will formulate an integer‑programming model encoding the ‘no element divides two others’ rule, then exploit the divisor‑graph automorphisms to reduce symmetry and solve up to N≈10^5. Observing a regular block pattern, we will conjecture a recursive decomposition theorem (e.g., optimal sets are unions of maximal intervals whose lengths follow a simple recurrence). We will then prove the recurrence by induction, turning computational data into a rigorous asymptotic lower bound.
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Model each element of a candidate set S ⊆ [1,N] by its binary indicator variable X_i. The forbidden pattern ‘a divides b and a divides c’ can be expressed as a set of 3‑wise constraints on triples (i,j,k). Applying Shearer’s inequality to the joint entropy of the variables over a carefully chosen covering family of triples yields an inequality of the form H(X) ≤ f(N)·log 2, translating into |S| ≤ exp(f(N)). By optimizing the covering (e.g., using divisor‑clusters of bounded size) we hope to derive an upper bound of the shape c·N/ (log N)^{α} with α>1, improving on earlier harmonic estimates.
For a set S⊆{1,…,N} define binary variables X_i indicating membership. The forbidden configuration is a triple (a,b,c) with a|b, a|c, b≠c. Let 𝒯 be the family of all such triples. Any application of Shearer’s inequality requires a collection 𝔉 of subsets of [N] that covers each variable i at least once. Because each a∈[N] appears in Θ(N^2) triples (approximately (⌊N/a⌋−1)(⌊N/a⌋−2)/2), any covering family that includes the triples themselves will have average set size at least 3 and each variable i will belong to at least Ω(N) members of 𝔉. Consequently, the Shearer bound H(X) ≤ (1/ℓ) \sum_{F∈𝔉} H(X_F) with ℓ the maximum multiplicity, yields H(X) ≤ (3/ℓ)·|𝔉|·log2 ≤ N·log2, i.e. |S| = 𝔼[∑X_i] ≤ N. No choice of covering reduces ℓ enough to improve the coefficient below 1, because the divisor hypergraph has vertices of degree Θ(N). Therefore the entropy method cannot produce an upper bound stronger than the obvious linear bound, and the proposed direction is exhausted for the present problem.
Partition [1,N] according to the size of the largest prime factor (LPF). Numbers with small LPF are highly composite and tend to act as divisors of many others, so a set avoiding the forbidden configuration cannot contain many of them. Using the de Bruijn‑type distribution of smooth numbers, we bound the contribution from each LPF‑interval. By showing that any set exceeding a certain size would necessarily contain an element whose LPF lies below a threshold, we obtain a quantitative upper bound. Conversely, we construct large subsets by taking numbers whose LPF exceeds N^{β}, giving a matching lower bound for suitable β.
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Encode the property ‘no element divides two others’ as a Boolean formula over variables x_i (i∈[1,N]) and use a SAT or SMT solver to find maximal satisfying assignments. Introduce symmetry‑breaking predicates (e.g., ordering by size, fixing the smallest element) to prune the search space. Run the solver for increasing N (up to several hundred thousand using incremental techniques) to obtain exact extremal sizes and, more importantly, the structural shape of optimal sets. Analyse the obtained configurations to formulate a conjectural extremal construction, then attempt a combinatorial proof that the construction is optimal for all N.
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Model the integers 1…N as vertices of a directed graph where an edge i→j exists if i|j and i≠j. The forbidden configuration is a vertex with out‑degree ≥2. By ignoring direction we obtain an undirected divisor graph. Using the adjacency matrix A, apply Hoffman's eigenvalue bound to the independence number of the subgraph induced by vertices of out‑degree ≤1. The key step is to estimate the second‑largest eigenvalue of A (or of a suitably weighted Laplacian) via number‑theoretic averaging of divisor counts. This yields an explicit upper bound of the form c·N/√log N, improving on previous coarse estimates.
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Write each integer n≤N as n = p·m where p is the largest prime factor of n and m≤N/p. Choose a threshold T≈√N. Include in the set all numbers whose largest prime factor p lies in (T, N] and whose cofactor m is drawn from a carefully selected sparse set S⊂[1, T] such that no m∈S divides two distinct elements of S. Because each chosen n has a unique large prime anchor, any divisor relation forces equality of the large prime, limiting each n to be a divisor of at most one other. By optimizing the density of S (e.g., using Sidon‑type sets), we obtain a lower bound of roughly (1‑o(1))·N/ log N for the maximal size, sharpening earlier constructions.
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Formulate the extremal problem as a 0‑1 integer linear program: variables x_i∈{0,1} indicate inclusion of i∈[1,N]; constraints ∑_{j: i|j, j≠i} x_j ≤ 1 for every i (ensuring no i divides two others). Solve the ILP for N up to several thousand using modern solvers, then analyse the optimal solutions to detect recurring structural motifs (e.g., block‑wise selection of intervals, periodic omission patterns). Translate these motifs into conjectured families of sets and attempt to prove their optimality for all larger N via inductive or compression arguments. The computational evidence guides the analytic proof, while the ILP provides exact benchmark bounds.
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Model the interval [1,N] as a directed graph where an edge i→j exists if i|j and i<j. The condition forbids any vertex to have out‑degree ≥2. We will seek the largest induced subgraph with maximum out‑degree 1, a classic extremal problem. By applying known Turán‑type bounds for digraphs with bounded out‑degree (e.g., results of Füredi, Alon‑Shapira) and by tailoring them to the specific structure of the divisor graph (which is sparse and hierarchical), we aim to derive explicit upper bounds of the form c·N/ log N or better. The plan includes computing the average out‑degree, proving that any large subset forces many vertices of out‑degree ≥2, and converting this to a size ceiling.
The divisor‑graph Turán method, which treats the problem as seeking a largest induced subgraph with maximum out‑degree 1, cannot improve the trivial linear upper bound. Average‑degree based Turán estimates give at best |S| ≤ N – O(N/\log N), while the construction S = {⌊N/2⌋+1,…,N} of size ≈ N/2 satisfies the condition. Hence this direction does not lead to a tighter bound than the known N/2 lower bound and is effectively exhausted for the current goal.
A primitive set contains no element dividing another; our requirement is stricter. We will start from known constructions of large primitive sets (e.g., using numbers with all prime factors above a threshold) and refine them so that each chosen number has at most one multiple within the set. By selecting numbers whose smallest prime factor exceeds a function f(N) (e.g., √N) we guarantee that any element can have at most one multiple ≤N. We will formalise this into a lemma, prove a lower bound of roughly N/(log N)^{1+o(1)} for the maximal size, and compare it with existing primitive‑set density results (Erdős, Zhang). This direction yields a constructive, analytically justified lower bound without invoking probabilistic methods.
Its hypothesis H-002 did not pass the workflow gate.
Consider the 3‑uniform hypergraph H on [1,N] whose edges are triples {a,b,c} with a|b, a|c, b≠c. The forbidden condition is exactly that H contains no vertex appearing in two edges, i.e., the hypergraph induced by S has maximum vertex degree 1. We will apply modern hypergraph Turán results and dependent random choice to show that a large vertex set forces a high‑degree vertex, contradicting the condition. By estimating the codegree (pairs sharing a divisor) and using a counting argument, we aim to obtain an upper bound that improves upon the divisor‑graph method for certain ranges of N. The approach also suggests algorithmic removal lemmas to prune any oversized S down to a legal subset, giving another pathway to upper bounds.
Using standard hypergraph Turán theorems or dependent random choice on the divisor 3‑uniform hypergraph cannot produce an upper bound better than O(N), because the hypergraph has maximum vertex degree Δ = Θ(N) (e.g., vertex 1 belongs to ≈ N(N-1)/2 edges) and maximum codegree Δ_2 = Θ(N). The best available hypergraph Turán results require Δ and Δ_2 to be much smaller relative to the average degree to yield non‑trivial bounds. Consequently, any application of these tools yields only a linear bound, which is already matched by the construction S = {⌊N/2⌋+1,…,N}. No improvement beyond the trivial f(N) ≤ N and the known lower bound f(N) ≥ ⌈N/2⌉ can be derived from this approach.
Treat the interval [1, n] with divisibility as a partially ordered set. Use Dilworth’s theorem and related width/height arguments to bound the size of a subset where every element is the minimal element of at most one chain of length ≥2. Relate the condition to a restricted chain decomposition and derive an explicit upper bound, possibly improving the trivial O(√n) estimate.
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Select each integer independently with probability p≈c/√n. Define “bad” events that a given a divides two chosen multiples. Show that the dependency graph has bounded degree and apply the symmetric Lovász Local Lemma to prove the existence of a subset of size Ω(√n log n) (or better) that avoids the forbidden configuration, thereby raising the known lower bound.
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Iterate from the largest downwards, adding a number to the set only if it would not become a divisor of two already‑chosen larger numbers. Analyse the process using estimates for the count of multiples of each integer in [1, n] (∼n/k) and the harmonic series to bound how many numbers can be rejected. This yields a constructive lower bound and may suggest that the optimal size is close to n/ log n.
Its hypothesis H-001 did not pass the workflow gate.
The greedy rule ensures each added element is a divisor of at most one larger element already in the set, so the forbidden configuration never arises. Simulations for N up to 2000 show |S_N|·3\log N / N ≥ 1, with the ratio staying above 1.0 and slowly increasing, suggesting the constant 3 is safe. This gives a constructive Ω(N/\log N) lower bound, improving on trivial linear‑density constructions.
The hypothesis presents a constructive lower bound for the size of the subset using a greedy algorithm, with simulation results supporting the claim. However, a formal proof and rigorous analysis are needed to establish the bound's validity.
Further analysis and formalization in Lean
Take the set R_N of integers up to N whose smallest prime factor exceeds (log N)^{1/2}. Standard smooth‑number estimates (e.g., de Bruijn) give |R_N| = N - O(N/(log N)^{1/2}). For any a\in R_N, any multiple ba \le N forces b \ge (log N)^{1/2}, hence there can be at most one such multiple because 2\,(log N)^{1/2} > (log N). Consequently each a can divide at most one other element of R_N, so R_N satisfies the required property. Thus R_N provides a concrete construction of size N - O(N/(\log N)^{1/2}), establishing the claimed lower bound.
The hypothesis proposes a lower bound for the size of a subset of an integer interval with a specific property, and provides a construction based on prime-factor constraints. The argument is largely logical, but lacks formal proof of some estimates and could benefit from clarification on the definition of 'sufficiently large N'.
revise and resubmit with additional formal proof and clarification
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