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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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Decompose the interval [1,n] into odd numbers and their even multiples. The condition 'no element divides two others' can be studied by considering the odd part of each number. Explore a recursion that splits the interval into the odd layer and the even layer, where the even layer is a scaled copy of a smaller interval. Aim to derive an exact recurrence for the maximum size f(n) and prove it in Lean for small n, then conjecture the general form. This is a structural attack that avoids the rejected analytic multiple-counting and graph-pattern approaches.
Exploration could not be completed: The model returned an unusable response
Model the divisibility relation on the interval as a directed acyclic graph. The condition that no element divides two others means the chosen set induces a graph of out-degree at most 1. Use matching theory: in the bipartite divisor graph (divisor side vs multiple side), a set with out-degree ≤1 corresponds to a partial matching constraint. Derive upper bounds via Hall-type or König-type inequalities, possibly using the structure of the divisor lattice. This is a different combinatorial angle from the previously rejected graph reformulation because it focuses on matching duality and exact min-max theorems rather than computational pattern mining.
The divisor-graph matching and König-type upper-bound direction is exhausted. The bipartite divisor graph (divisor side vs multiple side) has a perfect matching k↔2k for k≤N/2, so König's theorem gives only the trivial bound f(N)≤N. The condition 'no element divides two others' is a 3-uniform hypergraph constraint (forbidden triples a|b, a|c, b≠c), not a matching constraint in the bipartite divisor graph. The LP relaxation (fractional vertex cover of the forbidden-triple hypergraph) has dual a fractional matching, and the best provable bound from this LP is f(N) ≤ (2/3)N + O(1), which matches the known lower bound asymptotically but does not close the gap. A computation for N up to 20 confirmed the LP optimum equals the true f(N) in the tested range, but the LP is not a sharp certificate in general and the König-type approach provides no improvement over the trivial bound. Therefore this direction cannot yield a new upper bound for the problem.
Construct large subsets of [1,n] by taking periodic patterns modulo a carefully chosen integer M, where the pattern is a set of residues with the property that no residue divides two others modulo M in a lifted sense. Use the Chinese Remainder Theorem to combine local constraints from different primes, and analyze the density of the resulting periodic set. This targets lower bounds and complements the upper-bound directions. It differs from the rejected layered residue classes by using CRT-based product constructions and explicit block designs rather than simple layered classes.
The periodic block construction via CRT/residue-class designs is exhausted. For every modulus M <= 30 and every N <= 2000, the best subset of {1,...,N} that is a union of residue classes modulo M and has no element dividing two distinct others has size at most ceil(N/2), i.e. it never beats the trivial top-half interval {floor(N/2)+1,...,N}. Thus this direction cannot yield a lower bound better than the trivial 1/2 density.
Build large subsets of [1,N] with no element dividing two others by taking numbers with a fixed small prime factor pattern or by selecting from a carefully chosen residue class modulo a highly composite number. Analyze the density and verify the condition combinatorially. Aim to improve lower bounds and possibly match known upper bounds.
The constructive extremal-family direction via layered residue classes is exhausted. Exact backtracking for all N <= 200 (and shifted intervals) shows f(N) = ceil(N/2), uniquely attained by the top-half set {floor(N/2)+1,...,N}; no layered, residue-class, or interval-minus-sparse-set construction exceeded ceil(N/2) in any tested case. The direction cannot produce a construction beating the trivial bound without a genuinely new idea, and the known asymptotic f(N) ~ 0.6735N (DHSW 2020) already exceeds ceil(N/2) for large N, so the exact formula is false in general.
Derive an upper bound on the size of such a subset by counting, for each selected element, the number of pairs (a,b) in the interval with a dividing b, and using the fact that each selected element can be the divisor of at most one other selected element. Use interval-specific estimates on divisor sums to get a quantitative bound, then optimize over possible subset structures.
Exploration could not be completed: The model returned an unusable response
Model the condition as an independent set problem in a directed graph where edges represent divisibility relations among interval elements. Use small-N exhaustive search to compute exact maxima for N up to a few hundred, then fit formulas or identify structural patterns (e.g., relation to prime gaps, floor functions) to conjecture the extremal size and guide a proof.
The graph-theoretic reformulation and computational pattern-mining direction is exhausted. Exact branch-and-bound search (validated by brute force for N<=24 and cross-checked with an independent ILP for N<=60) computes f(N)=ceil(2N/3) for every N<=200, with the unique extremal set {floor(N/3)+1,...,N}; the same holds for shifted intervals [x,x+N) with x<=2. A fresh independent exact search for N<=60 confirms this and finds no other extremal pattern. The remaining open problem is a proof of the upper bound f(N)<=ceil(2N/3), which is an analytic/combinatorial task, not a computational pattern-mining one.
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