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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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46 SOURCES
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 This source is an abstract-only record of a paper on an exact weighted MAX-SAT solver (akmaxsat). It describes a branch-and-bound solver whose lower bound is computed via Max-SAT resolution plus detection of disjoint inconsistent subformulas, and introduces a propagation algorithm improving that detection, plus a lazy deletion data structure. Experiments show speedups on random instances with high clauses-to-variables ratio. The abstract contains no mathematics about divisibility, integer intervals, or the 'no element divides two others' extremal problem.
LITERATURE-01 read This source is an abstract-only record of a 2017 ergodic theory paper by Buczolich. It proves that the functions ω(n) and Ω(n) — the number of distinct prime factors of n, and the number of prime factors counted with multiplicity — are good weighting functions for the pointwise ergodic theorem in L¹. Specifically, for any ergodic dynamical system and any f ∈ L¹, the weighted averages with weights ω(n) or Ω(n) converge almost everywhere to the space average. This answers a question of Cuny and Weber, who had established the result only for Lᵖ with p > 1. The abstract contains no combinatorial statements about subsets of integer intervals, no divisibility conditions, and no bounds on set sizes.
LITERATURE-01 read The retrieved source is a supplementary file (metadata only) for a computational chemistry paper on testing exact upper bounds to exact exchange in density functional theory. It contains no mathematical content relevant to the problem of subset sizes in integer intervals with no element dividing two others. Only the title, DOI, and abstract-level metadata are available; no full text is held.
LITERATURE-01 read Only the metadata and abstract of Pollack's paper on divisor-weighted sums are available; the full text is not held by the lab. The abstract concerns divisor-weighted sums and related estimates, not the extremal problem of subsets of an integer interval with no element dividing two others. The abstract does not state or imply any bound for that problem.
LITERATURE-02 read This is a thesis metadata record (Shanise Walker, Iowa State University) on extremal graphs and poset theory. Only the title and abstract-level metadata are available; the full text is not held. The abstract is not provided, so the only supportable claim is that the thesis exists and concerns extremal problems in graphs and posets. It does not state any specific result about integer-interval subsets with no element dividing two others.
LITERATURE-01 read The retrieved source is only a metadata record for a table in a PeerJ Computer Science article about the complexity of brute-force and collision attacks on computation extractors. It contains no mathematical content relevant to the problem of bounding the size of a subset of an integer interval with no element dividing two others. The abstract is not even present; only the table title and DOI metadata are available.
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.
LITERATURE-02 read The retrieved source is a PeerJ supplemental data file (Log2 normalized count) with no full text held by the lab. It contains only metadata and an abstract, and the abstract is not provided. The source is a biological/genomics data supplement and has no mathematical content relevant to the problem of subset sizes in integer intervals with no element dividing two others.
LITERATURE-02 read This is a preprint in complex differential geometry about extremal functions and the Tian invariant on non-toric Fano manifolds, specifically the complex Grassmannian G_{n,2n+1}C. The abstract reports a proof of existence of an extremal function lowering all admissible functions with zero sup, invariant under a chosen automorphism group, to compute the Tian invariant. It has no stated connection to combinatorics, integer intervals, or divisibility conditions.
LITERATURE-01 read This is a chapter by Jeff Kahn on asymptotic results for hypergraph matching, covering, and coloring problems. Only the metadata and abstract are available; the full text is not held. The abstract indicates the chapter surveys asymptotic techniques for hypergraph matching/covering/coloring, but no specific results, bounds, or methods are stated in the available material.
LITERATURE-01 read This source is a figure caption for a Büchi automaton construction in a paper on ELTL (extended linear temporal logic) satisfiability checking. It describes Algorithm 2 for sat-set computation of ELTL formulae. It has no mathematical content relevant to divisibility in integer intervals.
LITERATURE-01 read Only the abstract/metadata of a paper on fast algorithms for finding extremal sets (maximal sets under a partial order, with applications to data mining) is available. The abstract concerns algorithmic techniques for enumerating extremal sets efficiently; it does not address divisibility relations on integer intervals or the specific extremal problem of subsets with no element dividing two others.
LITERATURE-01 read This is a thesis abstract on exact SAT and MaxSAT solving. It describes heuristic-guided formula simplification (winning SAT Competition 2022), an improved lower-bound estimation for branch-and-bound MaxSAT via an 'unlocking' mechanism (winning MaxSAT Evaluation 2024), and extensions of the MaxSAT Tableau proof system including signed MaxSAT. The abstract contains no mathematics about divisibility, integer intervals, or the specific extremal problem of subsets with no element dividing two others.
LITERATURE-02 read The abstract of Lebensold (1977) reports bounds on the size of the largest subset Q of {1,...,n} with no element dividing two others: as n grows, 0.6725n ≤ |Q| ≤ 0.6736n, with more accurate bounds achievable by additional computation. Only the abstract was available; the full paper was not read, so the method and proof details are not known from this source.
LITERATURE-01 read The retrieved source is a paper on simulating random vectors with given moments (Poirion, 2001). The abstract/metadata available concerns a numerical method for generating random vectors that match prescribed moment constraints. It has no connection to extremal combinatorics, divisibility relations, or subset sizes of integer intervals. The paper does not address the problem of bounding the size of a subset of an integer interval with no element dividing two others.
LITERATURE-01 read The source is a chapter on Integer Linear Programming (ILP) by Ping-Qi PAN, retrieved only as metadata and an abstract. The abstract-level content indicates the chapter covers ILP theory and methods, but no specific results, techniques, or details are available in the retrieved material.
LITERATURE-01 read The retrieved source is a PhD thesis on probabilistic analyses of combinatorial optimization problems on random shortest path metrics. It concerns random shortest path metrics and combinatorial optimization, not divisibility conditions on subsets of integer intervals. The abstract does not mention integer intervals, divisibility, or any related number-theoretic extremal problem.
LITERATURE-01 read This is a chapter metadata record from De Gruyter (1983), 'Infinite divisibility, generalities', with no abstract and no full text held by the lab. The only information available is the title, which concerns infinite divisibility in a probability/generalities context, not integer-interval subsets with divisibility-free conditions. The record provides no mathematical content, no results, and no techniques relevant to the problem of bounding the size of a subset of an integer interval with no element dividing two others.
LITERATURE-01 read This source is a figure caption from a PeerJ Computer Science article about metadata representation of DCAT Distributions using Turtle. It concerns RDF triples and metadata records, not mathematics. The abstract/metadata available supports only that the figure depicts a Turtle representation of a subset of triples from MetaRecord metadata pertaining to two DCAT Distributions. It contains no mathematical content, no results about integer intervals, divisibility, or subset sizes.
LITERATURE-01 read The source is a book chapter titled 'Short Circuit' from Springer New York, identified by DOI 10.1007/0-387-27160-0_21. Only the metadata and abstract are available; the full text is not held by the lab. The abstract is not provided in the retrieved metadata, so the chapter's content cannot be assessed. There is no evidence that this chapter addresses divisibility, integer intervals, or extremal subset problems.
LITERATURE-01 read The retrieved source is a 2000 statistics paper by Chakraborti on exact derivations of run length, average run length, and false alarm rate for Shewhart X-bar control charts, obtained by conditioning. Only the metadata and abstract are available; the full text is not held. The paper concerns statistical process control and has no mathematical content related to divisibility, integer intervals, or extremal combinatorics.
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Develop a counting argument that enumerates ordered triples (a,b,c) with a|b and a|c within {1,…,N}. By bounding the total number of such triples that can appear in any admissible set, derive an inequality relating |S| to N. The approach uses inclusion‑exclusion over prime powers and exploits the fact that each a can serve as a divisor for at most ⌊N/a⌋−1 other numbers, but the “no‑a‑divides‑two‑others” condition forces a strong reduction. The goal is to push the upper bound to O(N / log N) or better.
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Model the family of admissible subsets as independent sets in a 3‑uniform hypergraph H whose edges are triples {a,b,c} with a|b and a|c (b≠c). Apply the recent hypergraph container theorems to obtain a small family of “containers” each containing all independent sets. Analyzing the size and structure of these containers yields quantitative upper bounds on |S|. This line circumvents the earlier Turán‑type attempts by working with hypergraph sparsity rather than graph degree constraints.
Container theorems require bounded maximum vertex degree Δ and bounded maximum codegree Δ_2 relative to the average degree. In the divisor hypergraph, Δ ≈ N/2 (the element 1 belongs to ~N^2/2 edges) and Δ_2 is also Θ(N). These parameters make the standard container bound degenerate, yielding containers of size ≈ N and no improvement over the trivial linear bound. Thus the hypergraph container approach cannot produce a sub‑linear upper bound for f(N).
Partition {1,…,N} by residue modulo a carefully chosen integer M (e.g., a product of small primes). Show that if S is large, many of its elements must lie in the same residue class. Within a fixed class, use properties of multiplicative orders modulo M to prove that any two elements have a common divisor structure forcing a violation of the no‑double‑divisor rule unless the class size is bounded. Optimising M gives an explicit lower bound construction (e.g., using numbers with a fixed large prime factor) and improves the known lower bound for the extremal size.
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Model the set A 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. This is precisely a collection of stars with centre at each vertex of out‑degree ≤1. By limiting the total number of edges (which is bounded by the sum of floor(n/i)−1) we can apply Turán‑type results for graphs of bounded maximum out‑degree to obtain an upper bound on |A|. The direction will explore known extremal results for bipartite graphs and adapt them to the directed setting, yielding explicit inequalities such as |A| ≤ n / (log₂ n) + O(1).
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Select each integer in [1,n] independently with probability p and then delete elements that cause a violation (i.e., any element that becomes a divisor of two retained numbers). By optimizing p we can show the existence of subsets of size at least c·n / log n for a concrete constant c>0. The work will involve calculating the expected number of deletions using divisor counting functions and applying the Lovász Local Lemma to control dependencies. This yields a constructive lower bound that can be compared with the extremal upper bound.
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Consider the poset ( [1,n], | ) ordered by divisibility. Dilworth’s theorem states that the size of the largest antichain equals the minimum number of chains needed to cover the set. An element that divides two others forces a chain of length at least three. By partitioning [1,n] into the minimum number of chains and analysing how many elements can be kept from each chain while respecting the ‘at most one outgoing edge’ rule, we obtain refined upper bounds. In particular, the approach will examine the structure of chains formed by powers of 2 and other prime powers, leading to bounds of the form |A| ≤ n / (log₂ n) + O(log n).
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