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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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26 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 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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Partition the interval [1,N] according to the size of the largest prime factor (y‑smoothness). Within each class the divisor depth is bounded by O(log N / log y), limiting how many elements can share a common divisor. By choosing y as a slowly growing function of N, we can derive an upper bound of N / (log N)^{c} for some c>0, improving on trivial linear bounds. The approach will also explore whether a matching lower construction can be built from carefully chosen smooth numbers.
Its hypothesis H-003 did not pass the workflow gate.
Model the forbidden configuration (a divides b and a divides c) as a 3‑uniform hyperedge on vertices {a,b,c}. The problem becomes finding the largest independent set in this hypergraph. Apply known Turán‑type extremal results for 3‑uniform hypergraphs with bounded codegree (each pair appears in at most O(log N) edges) to obtain an upper bound of O(N / log N). Parallelly, adapt the dependent random choice technique to construct relatively large independent sets, giving improved lower bounds.
The hypergraph Turán method does not provide an upper bound better than O(N) for the maximum size of a subset of {1,…,N} with no element dividing two others; known constructions of size ⌈N/2⌉ show that any purported improvement would contradict these constructions. Hence this line of attack is exhausted for the goal of improving the asymptotic bound.
Implement an ILP formulation that enforces the divisor‑double‑free condition and solve it for N up to several thousand using modern solvers. Record the maximal cardinalities and the structure of optimal sets. Analyse the output to detect regularities (e.g., periodic gaps, preference for numbers with many distinct large prime factors) and formulate a conjectural recurrence or piecewise linear bound. Subsequent work will aim to prove the observed pattern analytically.
Exploration could not be completed: The model returned an unusable response
Treat the set {1,…,n} with the divisibility relation as a partially ordered set. Any admissible subset S has the property that each element belongs to at most one covering chain (an element can be a divisor of at most one larger element). Apply Dilworth’s theorem or Greene–Kleitman decomposition to bound the width of the poset, and translate this into an explicit upper bound on |S|, e.g. |S| ≤ n - ⌊log₂ n⌋. Formalise in Lean the definitions of divisor‑poset, chain, and prove the derived inequality.
Its hypothesis H-001 did not pass the workflow gate.
Select each integer in [1,n] independently with probability p, then delete any element that serves as a divisor of two retained numbers. Use the Lovász Local Lemma to show that for suitable p (≈1/2) the resulting set still has size Ω(n) while satisfying the no‑double‑divisor condition. Develop an algorithmic version (Moser‑Tardos) to produce an explicit set and prove its size bound. Formalise the random process and LLL application in Lean, yielding a lower‑bound lemma.
Its hypothesis H-002 did not pass the workflow gate.
Construct S by taking numbers in [1,n] whose smallest prime factor exceeds n^α for a chosen α (e.g., α=1/2). Such numbers can divide at most one other chosen number because any multiple would exceed n. Estimate the count of such ‘rough’ numbers via de Bruijn’s ψ‑function or Buchstab’s sieve, giving |S| = n - O(n/ log n). Refine the parameters to approach n - O(n/ (log n)^{c}) and compare with upper bounds. Formalise the rough‑number counting lemma and the resulting size estimate in Lean.
Exploration could not be completed: The model returned an unusable response
Consider the divisor poset ( {1,…,N}, | ). In any admissible set S each element can have at most one outgoing cover edge, so the induced subgraph is a disjoint union of chains. By Dilworth’s theorem the size of a minimum chain decomposition equals the width of the poset, i.e. the size of a largest antichain. The classic result of Erdős that the largest antichain in the divisor poset has size at most N / log₂ N gives a lower bound on the number of chains required. Since each chain contributes at most one element that is not a divisor of another element, at least ⌊log₂ N⌋ elements must be omitted to keep the out‑degree ≤1, yielding the stated inequality.
The hypothesis proposes a linear upper bound for the maximum cardinality of a subset S of an integer interval {1, ..., N} such that no element of S divides two distinct other elements of S. The argument relies on the divisor poset, Dilworth's theorem, and Erdős' result on the largest antichain. While the argument is mostly logically complete, there are missing assumptions and a need for formal proof to support some claims.
Revise and resubmit with additional formal proof and justification
A simple greedy algorithm that processes numbers from N down to 1 and adds a number whenever it does not cause any existing element to become a divisor of two retained numbers always produces a set of size at least ⌊N/2⌋+1 for N up to 200. This empirical evidence suggests a linear‑size construction may exist in general.
The hypothesis proposes a linear lower bound for divisor-double-free subsets, and the provided experiment code and report offer some empirical evidence to support this claim. However, the argument lacks formal proof and rigorous analysis of the greedy algorithm's performance for large N.
revise and resubmit with additional theoretical analysis
Partition [1,N] according to the size of the largest prime factor P⁺(n). Choose a smoothness threshold y = (log N)^α with α>1. Elements with P⁺(n)≤y are y‑smooth; any chain of divisibility among them has length at most O(log N / log y)=O(1/α). Hence each y‑smooth number can be the common divisor of at most O(1) pairs, limiting the number of y‑smooth elements that can appear in a divisor‑double‑free set to O(N / (log N)^{α}). The remaining numbers have a prime factor >y, so each such large prime can serve as a divisor for at most N / y numbers, again yielding a reduction by a factor of (log N)^{α}. Optimising α gives a concrete c>0 (e.g., α=2 yields c≈1). This heuristic suggests the claimed log‑power upper bound, while no construction is known to match it, indicating the bound may be non‑tight but improves on the linear bound.
The hypothesis proposes a log-power upper bound on the maximal cardinality of a subset with the property that no element divides two distinct other elements, using smooth-number decomposition. While the argument is partially convincing, it relies on heuristics and lacks explicit constructions and rigorous proofs, making it incomplete. The hypothesis runs into known barriers in number theory, particularly in establishing tight bounds on the number of smooth numbers and their divisibility properties.
revise and resubmit with additional rigor and explicit constructions
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