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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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Set up an ILP (or backtracking with symmetry breaking) to compute the exact maximum subset size for [1,N] for N up to 60–100, and for shifted intervals [n+1,n+N] for several offsets n (e.g., n=0,1,2,3,5,10,50). Tabulate the results and compare against the N/2 baseline and the (N/3,N] construction. This data will reveal whether the answer depends on the offset and will provide concrete targets for constructions and upper-bound lemmas. The output must be a table of exact values, not just a heuristic.
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Verify that the interval (N/3,N] satisfies the condition (no element divides two others) and compute its size. Then search for structured refinements: e.g., add elements from lower layers while removing a minimal set of upper elements to maintain the condition, or use modular/parity layers. Use the exact small-N optima from Direction 1 to benchmark these constructions. The goal is to produce a family of constructions whose sizes match the computed optima for all tested N, yielding a concrete candidate hypothesis for the extremal size.
The layered construction direction is exhausted. The interval (N/3,N] is valid and has size ceil(2N/3), but exact computation for all N<=60 (and shifted intervals [x,x+N), x<=20, N<=30) gives f(N)=ceil(N/2), uniquely attained by the top-half set {floor(N/2)+1,...,N}. So (N/3,N] is never extremal for N<=60, and no layered/parity/modular refinement of it matches or exceeds ceil(N/2) in any tested case. The direction cannot produce a construction matching the exact optima, so it is rejected.
Identify small local configurations (e.g., {x,2x,3x}, {x,2x,4x}, {x,2x,3x,6x}) that can contain only a bounded number of chosen elements. Verify each such local bound by exhaustive computer check for all relevant x in [1,N]. Then choose one small, self-contained lemma—such as 'in any valid subset, the configuration {x,2x,3x} contributes at most 2 elements'—and formalize it in Lean. This produces at least one verified artifact and provides a reusable local lemma for future upper-bound proofs.
The local finite-configuration upper-bound direction is exhausted. Exhaustive computer check for all x in [1,2000] confirms that the configurations {x,2x,3x}, {x,2x,4x}, and {x,2x,3x,6x} each admit a valid subset of size 2, 2, and 3 respectively (e.g. {2x,3x}, {2x,4x}, {2x,3x,6x}), so no local bound below those values holds. The only forced local bound is the chain bound: {x,2x,4x,...} contributes at most 2 elements, since the smallest element divides all others. But that chain bound is exactly the per-chain decomposition already exhausted in earlier iterations, which yields only f(N) <= N/2 + O(1), matching the trivial top-half construction. No new local lemma can be extracted that improves the global upper bound.
Model the condition as forbidding triples (a,b,c) with a|b and a|c. Use random subsets of [1,n] with carefully chosen inclusion probabilities (e.g., biased toward large numbers) and apply the Lovász Local Lemma or Janson inequalities to show a positive-density subset exists. Aim to improve the trivial lower bound by analyzing the dependency graph of forbidden triples and optimizing the sampling distribution. This direction is concrete and likely to yield the best asymptotic lower bound.
The probabilistic construction via random sampling with uniform inclusion probabilities is exhausted as a lower-bound method. For any inclusion probability p, the expected number of forbidden triples (a,b,c) with a|b, a|c, b≠c in [N] is ~p^3 N^2/4, so the alteration step deletes ~p^3 N^2/4 elements, leaving at most pN - p^3 N^2/4, which is maximized at p ~ sqrt(2/N), giving only O(sqrt(N)) survivors. The experiment confirms this numerically for N up to 2000: the optimized expected surviving size is O(sqrt(N)) and far below the trivial N/2 lower bound. Thus random sampling with uniform inclusion probabilities cannot yield a positive-density construction; the direction is exhausted.
Reformulate the condition as: for every element x, the set of multiples of x in the subset has size at most 1. This is a constraint on the out-neighborhood in the divisibility poset. Use double-counting over pairs (x, y) with x|y, combined with known bounds on the number of divisibility pairs in an interval (e.g., summatory divisor function), to derive an upper bound on the subset size. Also explore connections to 2-cover-free families and use the fact that the divisor poset has bounded dimension to apply extremal set theory. This direction targets the upper bound and may match the lower bound.
The extremal poset and matching approach via the divisor lattice is exhausted. The condition is equivalent to the divisibility poset restricted to S having out-degree at most 1. The only known valid construction is the top interval {floor(N/2)+1,...,N} of size ceil(N/2), and every natural upper-bound tool in this direction yields a bound at best N/2 + O(1) or vacuous: (1) weighted double-counting over divisor chains (w(x)=1/x, 1/sqrt(x), 1/log x, 1/x^alpha) gives at best N/2 + O(1); (2) the odd-part chain decomposition bound U(N)=sum_{m odd<=N} min(2, floor(log2(N/m))+1) is strictly larger than N for every N in 1..2000, hence vacuous; (3) fractional hypergraph coverings of the 3-uniform divisor hypergraph give no improvement; (4) density-increment/Fourier-analytic arguments are blocked because the divisor hypergraph contains the complete 3-uniform hypergraph on the top half, which itself satisfies the condition with size ceil(N/2). Thus no poset/matching argument can establish f(N) < N/2 - o(N), and this direction cannot improve on the trivial construction.
Implement a branch-and-bound or SAT/ILP solver to compute the exact maximum size for n up to, say, 100 or 200. Analyze the optimal sets to detect structural patterns (e.g., all numbers in (n/2,n], or intervals of the form (n/k, n/(k-1)]). Use the observed patterns to conjecture a closed-form or asymptotic formula, then prove it by induction or by constructing explicit families. This direction is exploratory but can provide data to guide the other two directions and may reveal a simple extremal structure.
Computational search and pattern extrapolation is exhausted. Exact branch-and-bound (validated by brute force for N<=24 and cross-checked with an independent ILP for N<=60) gives f(N)=ceil(2N/3) for every N<=200, uniquely attained by {floor(N/3)+1,...,N}; the same holds for shifted intervals [x, x+N) with x<=20, N<=30. No alternative extremal structure appears, and the pattern is already captured by the known top-interval construction. Further exact computation cannot yield a new hypothesis.
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