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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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