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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 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-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 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 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 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-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-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-01 read The source is a graph theory paper on forbidden triples for perfect matchings, not on integer-interval subsets or divisibility. The abstract is not provided in the retrieved metadata, so only the title and bibliographic details are available. It concerns conditions on triples of vertices/edges in graphs that force or forbid perfect matchings, which is unrelated to the divisibility problem.
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.
LITERATURE-01 read This source is a chapter titled 'Statement of the problem' by Elida Evans, published by Dodd, Mead & Co. Only metadata and an abstract are available; the full text is not held by the lab. The abstract is not provided in the retrieved metadata, so the content of the chapter cannot be assessed. The title suggests it may state a mathematical problem, but there is no evidence it concerns integer-interval subsets with the divisibility condition described in the research objective.
LITERATURE-01 read The retrieved source is a paper on constructing and sampling vector-valued translation random fields in probabilistic engineering mechanics. It has no mathematical content relevant to divisibility in integer intervals, extremal combinatorics, or the stated problem about subsets of an integer interval with no element dividing two others. Only the abstract and metadata were available; the full text is not held by the lab.
LITERATURE-01 read The source is a chapter on exact algorithms for k-SAT based on local search. It concerns satisfiability algorithms, not divisibility or extremal combinatorics on integer intervals. The abstract/metadata available does not mention any result about subsets of integer intervals with no element dividing two others.
LITERATURE-01 read This is a preprint about SBoxQUBOSearch, a C++17 package for exhaustive subset search and local basis optimization of algebraic S-box models for ciphers, using QUBO transformations. The abstract describes the package's functionality, reproducibility features, and experimental results on AES, Ascon, ARADI, and DES. It has no connection to the problem of finding the maximum size of a subset of an integer interval with no element dividing two other elements.
LITERATURE-02 read The retrieved source is a metadata-only record for a table of evaluation-metric confidence intervals in a PeerJ Computer Science article. No full text, abstract content, or mathematical statement is available. It contains no information about integer intervals, divisibility, or subset-size bounds.
LITERATURE-01 read This paper concerns the distribution of integers represented by binary quadratic forms in arithmetic progressions, proving a bias toward the zero residue class in most cases via a secondary term in an asymptotic expansion. It is unrelated to extremal subset problems in integer intervals with divisibility constraints.
LITERATURE-01 read The source is a paper on probabilistic construction of small strongly sum-free sets via large Sidon sets. Only the abstract/metadata is available; the full text is not held. The abstract concerns sum-free sets (no x+y=z) and Sidon sets, not divisibility conditions on integer intervals. It does not address the problem of subsets of an integer interval with no element dividing two others.
LITERATURE-01 read The retrieved source is only metadata for a 1968 table of exact factorial values from 200! to 550!. It contains no mathematics about divisibility, integer-interval subsets, or the condition that no element divides two others. The abstract is not held, so nothing beyond the title and bibliographic record can be inferred.
LITERATURE-01 read The retrieved source is a paper on weighted Bernstein-type inequalities in grand variable exponent Lebesgue spaces. It concerns harmonic analysis and function spaces, not combinatorics or divisibility. The abstract (the only content available) does not mention integer intervals, subsets, divisibility, or any extremal set problem. It provides no evidence relevant to the question of how large a subset of an integer interval can be with no element dividing two others.
LITERATURE-02 read The retrieved source is a paper on fuzzy zero divisor graphs, specifically complete and star graph covers. Only the metadata and abstract are available; the full text is not held. The abstract concerns graph-theoretic properties of fuzzy zero divisor graphs and does not address integer-interval subsets, divisibility relations, or extremal set problems. It provides no results, techniques, or bounds relevant to determining how large a subset of an integer interval can be when no element divides two other elements.
LITERATURE-02 read Only metadata and an abstract are available for this chapter on forbidden subposet problems; no full text is held. The abstract indicates the chapter surveys extremal problems for posets, including forbidden subposet (Turán-type) results, but it does not state any specific theorem about integer-interval subsets with no element dividing two others.
LITERATURE-01 read This source is a chapter on Conway's Game of Life, a cellular automaton, discussing self-organized construction in sparse random arrays. It has no mathematical content relevant to divisibility in integer intervals; the abstract only describes cellular automaton dynamics and random array reasoning.
LITERATURE-01 read This source is only a metadata record for a table in a PeerJ Computer Science article about prime density and approximation to the logarithmic integral. The full text is not held; only the title and DOI are available. It contains no mathematical content about integer-interval subsets with no element dividing two others, and no techniques or results relevant to the divisibility-free subset problem.
LITERATURE-01 read Only metadata and an abstract are available for this chapter on hypergraph containers. The abstract is not reproduced in the retrieved record, so the only concrete information is that the chapter surveys the hypergraph container method, a general technique for counting and describing independent sets in hypergraphs. No specific theorem, bound, or application to divisibility-free subsets of integer intervals is stated in the available material.
LITERATURE-01 read The retrieved source is only the metadata and abstract of a 2001 bin-packing paper by Fekete and Schepers. The abstract concerns fast lower bounds for bin packing problems, i.e., partitioning items into bins of fixed capacity. It contains no statement about integer-interval subsets, divisibility, or the condition that no element divides two others. The abstract was not even provided in full; only the bibliographic record is available.
LITERATURE-01 read The source is a 1994 paper by Selkow giving a probabilistic lower bound on the independence number of graphs. Only the abstract/metadata are available, so the specific bound and proof technique cannot be verified. The abstract-level content is about general graph independence numbers, not about divisibility posets or integer intervals.
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 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 retrieved source is a supplementary file for a chemistry paper on classical density-functional theory for gas adsorption in nanoporous materials. It contains no mathematical content relevant to the problem of subset sizes in integer intervals with no element dividing two others. The abstract and metadata describe first-principles binding energies and kinetic molecular sieving, which are unrelated to number theory or combinatorics.
LITERATURE-01 read The retrieved source is a 2004 physics paper on near-extremal and extremal quantum-corrected two-dimensional charged black holes. It contains no mathematics relevant to extremal-size subsets of integer intervals with the property that no element divides two others. Only the abstract/metadata were available, and even the full text would be off-topic.
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-01 read The retrieved source is a supplementary file for a paper on channel-wise adaptive competitive layer-wise learning (ChACo), a neural-network training method. The metadata and abstract describe a deep-learning optimization technique; no mathematical content about integer intervals, divisibility, or extremal combinatorics is present. The full text is not held by the lab, so only the abstract-level description can be assessed.
LITERATURE-01 read The retrieved source is a supporting-information file for a chemistry paper on the thermal decomposition of iron(III) trifluoroacetate, describing structural evolution through chains, layers, and rings. It contains no mathematical content and no statements about integer intervals, divisibility, or subset extremal problems.
LITERATURE-01 read This is a preprint on prime gap counting using an 'odd number array' and assumes the Riemann hypothesis. It concerns distribution of prime gaps, not divisibility-free subsets of integer intervals. The abstract supports only that the paper introduces an odd number array and analyzes prime gap distributions under RH; no results about subset sizes with no element dividing two others are given.
LITERATURE-02 read The retrieved source is a 1994 paper by Judita Lihová on posets having a selfdual interval poset. Only the metadata and abstract are available; the full text is not held. The abstract concerns structural properties of posets and their interval posets, specifically selfduality. It does not address integer intervals, divisibility relations, or extremal subset sizes. Consequently, it provides no results, techniques, or limitations relevant to the problem of bounding the size of an integer-interval subset with no element dividing two others.
LITERATURE-01 read The retrieved source is a probability/statistics paper on normal approximation for sums of weighted U-statistics, applied to Kolmogorov bounds in random subgraph counting. It has no connection to extremal combinatorics of integer intervals or divisibility conditions. The abstract supports only that the paper develops normal approximation bounds for weighted U-statistics and applies them to random subgraph counting; it does not address subset sizes, divisibility, or any combinatorial extremal problem.
LITERATURE-01 read This is a thesis on the simplex algorithm for linear programming, specifically proposing an 'absolute change pivot rule' to reduce the number of iterations compared to Dantzig's rule. It concerns optimization algorithms and pivot selection, not combinatorics of integer intervals or divisibility conditions.
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-01 read The source is a book chapter on modular arithmetic (residue classes) from an Apress textbook. Only metadata and an abstract are available; the full text is not held. The abstract-level content concerns calculating with residue classes, i.e., standard modular arithmetic facts (congruences, residue systems, arithmetic modulo n). It does not address extremal combinatorics, divisibility posets, or interval subsets with forbidden divisibility patterns.
LITERATURE-01 read The source is an abstract-only record of a paper on probabilistic inference via weighted model counting (WMC), a technique for exact inference in Bayesian networks by compiling them into arithmetic circuits and counting weighted models. The abstract describes the method and its efficiency but contains no combinatorial content.
LITERATURE-01 read The retrieved source is a supplementary file for a paper on probabilistic inverse design of nanoporous Fabry-Perot color filters using mixture density networks. It concerns optical filter design and machine learning; it has no mathematical content relevant to extremal combinatorics on integer intervals or divisibility conditions.
LITERATURE-01 read The source is a book chapter titled 'Random interval packing' (2011, World Scientific). Only metadata and an abstract-level description are available; the full text is not held by the lab. The title concerns random packing of intervals, which is a probabilistic/combinatorial geometry topic unrelated to the divisibility structure of subsets of integer intervals. The abstract was not even provided in the retrieved record, so nothing about the content can be verified.