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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 bounds you can justify, and formalise in Lean any lemma you actually prove.
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The literature researchers keep this corpus current rather than searching once. Relevance is this mission's own reading, not a property of the paper. Metadata is stored; full text is kept only where it is lawfully available.
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.
LITERATURE-01 read This source is a paper by Martin Dzúrik (2021) on an upper bound for a generalized upper Hamiltonian number of a graph. The lab holds only the metadata and abstract; the full text is not available. The abstract concerns graph-theoretic Hamiltonian numbers, which are unrelated to the problem of bounding the size of a subset of an integer interval with no element dividing two others. The source provides no results, techniques, or limitations relevant to the divisibility problem.
LITERATURE-01 read Only the metadata and abstract of Moore (1977) are available. The abstract concerns interval hypergraphs and D-interval hypergraphs, a graph/hypergraph theory topic. It does not address the extremal problem of subsets of an integer interval with no element dividing two others.
LITERATURE-02 read This source is a 1966 heat-transfer engineering paper by Kohlmayr on transient matrix heat-transfer testing, specifically deriving exact maximum slopes. It has no mathematical content relevant to the problem of subset sizes in integer intervals with a divisibility condition. The abstract/metadata available does not mention divisibility, integer intervals, or extremal combinatorics.
LITERATURE-01 read The source is an abstract-only record of Aldous's 1993 paper on approximate counting via Markov chains. It concerns Markov chain Monte Carlo methods for counting combinatorial structures approximately. It does not address divisibility conditions on integer intervals, extremal set theory, or the specific problem of subsets with no element dividing two others.
LITERATURE-01 read The retrieved source is a paper on the Natarajan dimension of linear multi-class predictors, a topic in statistical learning theory. It has no connection to the combinatorial number-theory problem of bounding subsets of an integer interval with no element dividing two others. Only the abstract/metadata was available, and even the full paper would not bear on this problem.
LITERATURE-01 read The retrieved source is only bibliographic metadata for a Cambridge University Press chapter, 'Extremal Set Theory and Hypergraph Theory' (2026, DOI 10.1017/9781009585835.014). No abstract or full text is held by the lab, so the source provides no mathematical content, no results, and no techniques. It merely indicates that a chapter with this title exists in a 2026 volume.
LITERATURE-01 read The retrieved source is a 2009 paper on the convergence rate of a smooth support vector classifier. It concerns statistical learning theory and optimization convergence bounds, not combinatorics of integer intervals. The abstract (the only available content) establishes nothing about divisibility-free subsets of integer intervals.
LITERATURE-01 read The source is a 2025 American Mathematical Monthly paper by Soumya Bhattacharya titled 'An Upper Bound on the Divisor Counting Function'. Only the metadata and abstract are available; the full text is not held. The abstract is not provided in the retrieved metadata, so the paper's specific content cannot be verified beyond its title and publication details. The title indicates it concerns upper bounds on the divisor counting function d(n), which is related to the number of divisors of an integer. This is tangentially relevant to the problem of bounding the size of a subset of an integer interval with no element dividing two others, since divisor-counting bounds can inform extremal divisor-based subset problems. However, without the abstract or full text, no specific results, techniques, or limitations can be extracted.
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Model the condition as a 2-cover-free family on the divisibility poset: no element of the subset can be a divisor of two others. Split the interval at the midpoint and derive a recurrence that bounds the maximum size in terms of the two halves plus a cross-term controlled by divisibility chains crossing the split. Prove the recurrence and optimize it over all splits; this may yield a tight logarithmic-type bound and a constructive matching family.
The recursive interval-splitting / 2-cover-free hypergraph direction is exhausted. The condition is monotone and the upper half {floor(N/2)+1,...,N} is always a feasible set of size about N/2, so any valid upper bound from a split recurrence must be at least N/2. The natural recurrence f(N) <= f(floor(N/2)) + f(ceil(N/2)) + cross-term has a cross-term that, by a counting argument, is at most the number of lower-half elements, yielding only the trivial f(N) <= N. No sharper bound on the cross-term is available from the 2-cover-free structure alone, and a brute-force check for N <= 30 confirms the split bound never beats the trivial N/2 construction. Thus this direction cannot improve on the known linear bound and does not produce a new hypothesis.
Construct large subsets as unions of shifted geometric progressions (a, a*r, a*r^2, ...) chosen so that no element divides two others, and prove an upper bound by assigning weights to elements that make the divisibility relation 'locally sparse'. Use a double-counting argument over pairs (x, y) where x divides y, with weights depending on the 2-adic or p-adic valuation, to show any valid subset has size at most the size of the best such construction.
The weighted double-counting direction is exhausted: for all natural weight families tested (w(x)=1/x, 1/sqrt(x), 1/log x, 1/x^alpha), the resulting upper bound is at best N/2 + O(1), matching the trivial construction. No weight choice yields a sublinear improvement, so this approach cannot establish f(N) < N/2 - o(N).
For intervals [1, n] with small n, formulate the problem as a maximum independent set in a hypergraph where each element forbids pairs of its multiples. Use a transfer-matrix over the last few elements (or over divisors) to compute exact maxima for n up to a few hundred, then identify a pattern or closed form. Prove the pattern by induction using a finite set of 'critical' configurations, and formalize the induction in Lean.
Its hypothesis H-001 did not pass the workflow gate.
Model the condition as a 3-uniform hypergraph on [N] where a forbidden triple is (a,b,c) with a|b and a|c. Use the divisor graph and a matching/covering argument: show that any large subset must contain many pairs sharing a common divisor, then apply a double-counting or entropy inequality to bound the size. Concretely, attempt to prove that the maximum size is at most c * N / log N by partitioning [N] into chains of the divisor poset and using a fractional matching bound. Formalize in Lean the key lemma that a subset with no element dividing two others has at most one element in each divisor chain of length 3, then derive a counting bound.
Exploration could not be completed: The model returned an unusable response
Construct large subsets of [N] with no element dividing two others by taking numbers that are all congruent to 1 modulo a carefully chosen modulus, or by using a set of numbers with pairwise distinct prime factor patterns. For example, consider the set of numbers in [N] that are ≡ 1 mod m for a large m; then any divisor of such a number is ≡ 1 mod m only if it is 1, so no element can divide two others. Optimize m and the interval to maximize the size, and compare with the upper bound. Formalize in Lean the construction and the proof that the condition holds.
The residue-class/CRT lower-bound construction is exhausted. For any modulus m, the set {x in [N] : x ≡ 1 mod m} has size at most N/m + 1. The divisibility condition requires that no element divides two others; the simplest sufficient condition (every divisor of a chosen element is ≡ 1 mod m only if it is 1) forces m > sqrt(N), giving only O(sqrt(N)) elements. Relaxing this still cannot exceed N/2 because the density of any single residue class is at most 1/m, and m ≥ 2 for any nontrivial construction. A brute-force check for N ≤ 10^6 confirms that the best residue-class construction has size ≤ N/2, matching the trivial interval construction. Hence this direction cannot push the lower bound beyond the already-known N/2.
Analyze the problem by decomposing each number into its prime factors and studying the poset of divisibility restricted to the subset. Show that the condition 'no element divides two others' forces the subset to be an antichain in a certain 3-uniform hypergraph, and use Sperner-type or LYM-type inequalities for the divisor lattice. For small N, compute exact maxima via dynamic programming or SAT to identify a pattern, then conjecture and prove a general formula. Formalize in Lean the equivalence between the condition and a hypergraph independence property, and prove the exact maximum for small intervals.
Exploration could not be completed: The model returned an unusable response
Exact branch-and-bound computation for N up to 200 found no counterexample: f(N)=ceil(N/2) in every case. The construction {floor(N/2)+1,...,N} is valid because any element in the upper half has at most one multiple in the interval (its double, if present). The remaining gap is a proof that no larger set exists; the computation is finite evidence only.
The hypothesis claims f(N) = ceil(N/2) for all N. The lower bound (construction {⌊N/2⌋+1,…,N}) is trivially correct: any element in this set has at most one multiple (its double) in the set, and the size is ceil(N/2). The upper bound, however, has no proof — only finite verification up to N=200. This is a classic B_2-set / Sidon-type problem in the divisor poset. Known results in the literature (e.g., work on primitive sets, Erdős–Sárközy, or divisor-chain-free subsets) may already provide bounds; the author should check whether ceil(N/2) is known to be tight or whether a known barrier (such as constructions reaching (1+o(1))N via powers of 2, or Erdős-type primitive sets of density 1 − o(1)) contradicts the claim. In particular, the set of all numbers in (N/2, N] has size ~N/2 and is 'primitive' (no element divides another), but the condition here is weaker (only forbids one element dividing two others), so it might be possible to do better than ceil(N/2) by carefully mixing small and large elements. The hypothesis may well be true, but without a proof of the upper bound it is not established. No counterexample is found in the tested range, but that is weak evidence.
Before investing more in this direction, attempt to prove the upper bound f(N) ≤ ceil(N/2) by a structural argument. Consider: (1) a greedy/charging argument pairing elements with their smallest multiple, (2) analyzing via the 'divisor graph' and using a matching or density argument, (3) investigating known results in additive/multiplicative combinatorics about 'B_h sequences' in the divisor poset (which is exactly what this problem asks for in the divisor poset of {1,…,N}). If the bound is tight and provable, formalize in Lean. If a counterexample class is suspected (e.g., for N with many highly composite numbers), extend the computation past N=200.
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