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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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DIRECTION
HYPOTHESIS
REVIEW
PROOF
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Translate the condition ‘no element divides two others’ into a bound on the average order of the divisor function τ(n) over S. Use known estimates for ∑_{n≤N} τ(n) and for the distribution of numbers with few divisors to derive an upper bound of the form |S| ≤ c·N / log N (or a refined logarithmic factor). Complement this with a lower bound obtained by selecting numbers with a prescribed large prime factor, whose divisor counts are small, giving a constructive set of size ≈ N / (log N)^{1+o(1)}. The approach yields asymptotically tight bounds up to poly‑logarithmic factors.
Its hypothesis H-001 did not pass the workflow gate.
Partition the interval [1,N] according to the size of the largest prime factor (LPF). Show that any element whose LPF exceeds N^{α} (for a fixed α<1) can be used at most once in a forbidden triple, because any multiple would exceed N. Construct S by taking all numbers with LPF > N^{α}, giving a lower bound of roughly N·(1-α). For the upper bound, prove that if S is too large then many elements must share a common small prime factor, forcing a divisibility pair. Use results on smooth numbers (de Bruijn, Hildebrand‑Tenenbaum) to quantify the proportion of numbers without a large prime factor and obtain a bound of the form |S| ≤ N·(1- c·α)+o(N). This line gives a linear‑in‑N bound with an explicit constant derived from smooth‑number densities.
Its hypothesis H-002 did not pass the workflow gate.
Define a 3‑uniform hypergraph H_N on vertex set [1,N] where {a,b,c} forms a hyperedge iff a|b and a|c with b≠c. The problem asks for the maximum size of an independent set in H_N. Apply Turán‑type results for 3‑uniform hypergraphs with bounded codegree, using the known bound that the extremal number ex_3(N, F) ≤ O(N^{2‑1/Δ}) where Δ is the maximum degree of the forbidden configuration. Estimate the codegree in H_N via divisor counting to show Δ≈log N, yielding an upper bound |S| = O(N^{2‑1/ log N})≈ O(N/ log N). Complement this with the hypergraph regularity lemma to obtain a structural decomposition of S, leading to a more precise bound and possibly a stability statement. This direction frames the problem in extremal combinatorics and leverages powerful generic tools.
Applying generic 3‑uniform hypergraph Turán bounds to the divisor hypergraph yields only the trivial linear upper bound (or even a weaker super‑linear bound), and contradicts known constructions of size ≈ N/2. Hence this hypergraph Turán line of attack is exhausted for obtaining meaningful bounds.
Treat the set \([1,N]\) with the divisibility relation as a partially ordered set. The condition “no element divides two others” is equivalent to requiring that every element lies in at most one comparable pair, i.e. the selected set forms a family where each element is the unique minimal element of at most one chain of length three. Apply Dilworth’s theorem, Sperner’s theorem, and known bounds on the size of antichains in the divisor lattice to derive upper bounds of order \(N/\log N\) (or sharper) and to relate the problem to chain decompositions and the height of the poset.
Exploration could not be completed: The model returned an unusable response
Use the probabilistic method to build large subsets with the desired property. Start with a random subset of \([1,N]\) where each integer is chosen with probability p. Estimate the expected number of “bad” triples (a|b and a|c). Choose p to keep this expectation below 1, then apply deletion or the Lovász Local Lemma to eliminate all violations, yielding explicit lower‑bound constructions. Explore how the resulting bounds compare with the antichain approach.
Exploration could not be completed: The model returned an unusable response
Implement exhaustive search or integer‑programming models for intervals up to several thousand to compute exact maximal sizes. Analyse the structures of optimal sets (e.g., clustering around primes, powers of two, etc.). Use the data to formulate conjectural formulas or recurrence relations, and test greedy heuristics (adding the smallest admissible element) for near‑optimal performance. This empirical work can suggest new combinatorial lemmas to be proved formally in Lean.
Exploration could not be completed: The model returned an unusable response
The divisor function τ(n) has average order log n, so a set where each element can be a divisor of at most one other element cannot be too dense; summing τ over the set gives a bound of order |S|·log N ≤ 2N, yielding |S|=O(N/log N). Conversely, taking numbers with a prime factor > N^{1/2} (or more generally > N^{1-ε}) gives a construction of size ≍ N/ log N, showing the bound is tight up to constants.
The hypothesis provides a plausible argument for the asymptotic size of divisibility-restricted subsets, but lacks formal proof and rigorous analysis of the lower bound construction.
Revise and resubmit with additional formal proof and analysis
A simple construction shows a lower bound of ⌈N/2⌉: all numbers larger than N/2 have no multiples within the interval, so no element can divide two others. For the upper bound, partition S into A={x∈S : x≤N/2} and B=S\A. The map φ:A→{2x : x∈A} is injective and its image lies in {1,…,N}. If |A|>⌊N/2⌋ then φ(A) would exceed N, impossible. Hence |A|≤⌊N/2⌋. Elements of B are already >N/2, so |B|≤⌈N/2⌉. Adding gives |S|≤⌈N/2⌉+⌊N/2⌋ = N, but the condition that any x∈A can have at most one multiple in S forces at least one element of φ(A) to be excluded from B, tightening the total to ⌈N/2⌉. This argument aligns with the intuition that numbers with a large prime factor (greater than N^{1/2}) cannot appear together with many smaller numbers without creating a divisor that splits into two multiples.
The hypothesis provides a plausible upper bound for the size of the subset, and the proposed construction achieves the lower bound. However, the argument relies on some intuitive assumptions and lacks formal proof for certain steps.
proceed with formalization in Lean and further analysis
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