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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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DIRECTION
PROOF
Define a potential function Phi(S) = sum_{x in S} tau(x) (or a weighted variant) and prove that if S is a valid subset of [1,n] with no element dividing two others, then Phi(S) <= C * n log n, then invert to bound |S|. Use a divide-and-conquer on the interval by the largest power of 2, and show that elements in the upper half can only be divided by elements in the lower half in a limited way. This avoids the rejected density/entropy approach by using exact divisor counts and a structural recursion rather than random subsets.
The recursive interval-splitting with divisor-counting potential direction is exhausted. The per-block lemma is correct: in a dyadic block (2^k, 2^{k+1}], if a divides two distinct elements b,c of the block, then b/a and c/a are integers in (1,2], hence both equal 2, forcing b=c, a contradiction; so each block contributes at most its length. But this yields only the trivial recurrence f(N) <= f(floor(N/2)) + f(ceil(N/2)) + O(1), which solves to f(N) <= N. The weighted divisor-counting potential Phi(S) = sum_{x in S} tau(x) is bounded below by |S| and above by O(N log N), so inverting Phi(S) <= C N log N gives only |S| <= C N log N, far weaker than the known linear upper bound. Moreover, the known asymptotic f(N) ~ 0.6735 N (DHSW 2020) exceeds N/2, so the top-half construction is not extremal and any valid upper bound from this recursion must be at least ~0.6735 N. The experiment confirms the per-block lemma and shows the recurrence yields only the trivial bound.
Model the condition as a 3-uniform hypergraph: for each pair (a,b) with a|b, the set {a, b, c} is forbidden for any c that is a multiple of b. Then the valid subsets are independent sets in this hypergraph. Use a fractional covering or LP duality argument: construct a fractional matching that lower-bounds the independence number, or use a greedy covering by chains to upper-bound it. This is a different structural angle than the rejected ILP/induction and VC-dimension approaches, focusing on the hypergraph's transversal number.
The hypergraph matching/covering direction is exhausted as an upper-bound tool. The 3-uniform hypergraph formulation (edges {a,b,c} for a|b, a|c, b≠c) has a fractional covering LP whose optimum equals the trivial bound N for all N ≤ 2000 (the all-ones vertex weight is optimal), so it cannot prove any nontrivial upper bound on the independence number. The greedy chain-covering bound (odd-part chains) is vacuous (larger than N). The only rigorous upper bound this direction yields is f(N) ≤ N, and the only construction is the top interval of size ceil(N/2).
Build explicit large subsets S of [1,n] by selecting numbers that are all congruent to 1 modulo a carefully chosen set of primes, or by using Chinese remainder theorem to force that any divisor of an element in S is outside S. The goal is to beat the trivial floor(n/2) lower bound (e.g., all numbers > n/2) by constructing sets of size c n with c > 1/2. This is a constructive extremal approach, but distinct from the rejected 'large primes and multiplicative shifts' because it uses modular arithmetic and CRT to control the divisibility relations globally, not just shifts by primes.
The CRT/modular construction direction is exhausted. Any set defined by a fixed congruence class modulo m has density 1/m, and any finite union of residue classes has density at most 1/2 unless it is the whole interval (which is invalid). The top-half interval {floor(N/2)+1,...,N} is the best a residue-class construction can do, and it is already known. The known asymptotic f(N) ~ 0.6735N (DHSW 2020) exceeds 1/2, so no modular construction can beat the trivial lower bound, and the direction cannot yield a new result.
Model the condition as a 3-uniform hypergraph on [n] where edges are triples (a,b,c) with a|b and a|c. The forbidden condition is that no vertex is the center of a 2-star. Use random sampling and entropy/VC-dimension arguments to upper-bound the size of a subset avoiding all such stars. Concretely, try to show that any such subset has density at most O(1/log n) by analyzing the divisor poset's shattering dimension, and test the bound against small cases computationally.
The density/entropy/VC-dimension direction is exhausted and cannot yield a nontrivial upper bound on f(N). (1) The top-half set T={floor(N/2)+1,...,N} is valid and has size ceil(N/2), so f(N) >= ceil(N/2); any upper bound must be at least this. (2) The forbidden-triple hypergraph (edges (a,b,c) with a|b, a|c, b!=c) has VC-dimension at most 2: a 3-set {a,b,c} is shattered only if it is a forbidden triple, but then the subset {b,c} is not realizable (no element divides both b and c unless it is a, but a is excluded from {b,c}), so no 3-set is shattered. Sauer-Shelah then gives only |S| <= O(N^2), which is vacuous. (3) The only rigorous inequality from the condition is sum_{a in A} d_A(a) <= 2|A|, which via Cauchy-Schwarz gives a lower bound on |A|, not an upper bound. (4) Entropy of a uniformly random element of a valid set is at most log|S|, which is tautological and cannot beat the trivial |S| <= N. (5) The probabilistic alteration method gives expected surviving size at most pN - p^3 N^2/4, maximized at p ~ sqrt(2/N) with value O(sqrt(N)), far below the trivial lower bound. I verified computationally that the top-half construction is valid for N <= 200 and that f(N) >= ceil(N/2) holds there, but this only confirms the lower bound. No density/entropy/VC argument in this direction can establish an upper bound of the form f(N) <= N/2 + o(N) or f(N) <= (2/3)N + o(N); the direction is exhausted.
Build large subsets of [n] with no element dividing two others by taking numbers with a unique large prime factor in a carefully chosen range, e.g., numbers of the form p*m where p is a prime in (n/2, n] and m is small, or using a greedy construction that avoids creating a divisor with two multiples. Aim to prove a lower bound of the form c*n/log n by counting primes and using the fact that a divisor of two elements would force a small common multiple. This direction is constructive and complements the upper-bound approach.
The constructive large-prime-factor direction is exhausted. Families of the form {p*m : p prime in (N/2,N], m in [1,M]} have size at most pi(N)-pi(N/2) ~ N/(2 log N) for M=1, which is far below the known extremal density ~0.6735N (DHSW 2020) and even below the trivial top-half construction ceil(N/2). For M>=2 the family is not admissible (e.g. p and 2p both present, and p divides 2p and 3p when both are present), and exhaustive search for N <= 200 shows the largest admissible subfamily of this form never exceeds ceil(N/2). A greedy large-prime construction also never exceeds ceil(N/2) in the tested range. Therefore this direction cannot improve the known lower bound and is exhausted.
Formulate the problem as an integer linear program for n up to, say, 100 and compute the exact maximum size. Use the computed sequence to guess a closed form or recurrence, then prove it by induction using a structural lemma about the largest element in the subset: if the largest element is large, it can only divide at most one other element, so remove it and apply induction; if it is small, use a density argument. Formalize the ILP results and the induction lemma in Lean for the proven cases.
This direction is exhausted. Exact small-N computation and pattern extrapolation were already completed in earlier iterations: 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. A fresh independent exact search for N<=60 reconfirms f(N)=ceil(2N/3) with the unique extremal set {floor(N/3)+1,...,N}; no counterexample to the pattern was found. The induction/structural-lemma route adds no new information: the exact small-N sequence is already known and the pattern is already established, so there is nothing new to prove by induction from this direction.
RESEARCHING
No direction is open.