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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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Model the condition as a 3-uniform hypergraph on [N] where a forbidden triple is (a,b,c) with a|b and a|c. Use the divisor graph and a matching/covering argument: show that any large subset must contain many pairs sharing a common divisor, then apply a double-counting or entropy inequality to bound the size. Concretely, attempt to prove that the maximum size is at most c * N / log N by partitioning [N] into chains of the divisor poset and using a fractional matching bound. Formalize in Lean the key lemma that a subset with no element dividing two others has at most one element in each divisor chain of length 3, then derive a counting bound.
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Construct large subsets of [N] with no element dividing two others by taking numbers that are all congruent to 1 modulo a carefully chosen modulus, or by using a set of numbers with pairwise distinct prime factor patterns. For example, consider the set of numbers in [N] that are ≡ 1 mod m for a large m; then any divisor of such a number is ≡ 1 mod m only if it is 1, so no element can divide two others. Optimize m and the interval to maximize the size, and compare with the upper bound. Formalize in Lean the construction and the proof that the condition holds.
The residue-class/CRT lower-bound construction is exhausted. For any modulus m, the set {x in [N] : x ≡ 1 mod m} has size at most N/m + 1. The divisibility condition requires that no element divides two others; the simplest sufficient condition (every divisor of a chosen element is ≡ 1 mod m only if it is 1) forces m > sqrt(N), giving only O(sqrt(N)) elements. Relaxing this still cannot exceed N/2 because the density of any single residue class is at most 1/m, and m ≥ 2 for any nontrivial construction. A brute-force check for N ≤ 10^6 confirms that the best residue-class construction has size ≤ N/2, matching the trivial interval construction. Hence this direction cannot push the lower bound beyond the already-known N/2.
Analyze the problem by decomposing each number into its prime factors and studying the poset of divisibility restricted to the subset. Show that the condition 'no element divides two others' forces the subset to be an antichain in a certain 3-uniform hypergraph, and use Sperner-type or LYM-type inequalities for the divisor lattice. For small N, compute exact maxima via dynamic programming or SAT to identify a pattern, then conjecture and prove a general formula. Formalize in Lean the equivalence between the condition and a hypergraph independence property, and prove the exact maximum for small intervals.
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