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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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Develop a counting argument that enumerates ordered triples (a,b,c) with a|b and a|c within {1,…,N}. By bounding the total number of such triples that can appear in any admissible set, derive an inequality relating |S| to N. The approach uses inclusion‑exclusion over prime powers and exploits the fact that each a can serve as a divisor for at most ⌊N/a⌋−1 other numbers, but the “no‑a‑divides‑two‑others” condition forces a strong reduction. The goal is to push the upper bound to O(N / log N) or better.
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Model the family of admissible subsets as independent sets in a 3‑uniform hypergraph H whose edges are triples {a,b,c} with a|b and a|c (b≠c). Apply the recent hypergraph container theorems to obtain a small family of “containers” each containing all independent sets. Analyzing the size and structure of these containers yields quantitative upper bounds on |S|. This line circumvents the earlier Turán‑type attempts by working with hypergraph sparsity rather than graph degree constraints.
Container theorems require bounded maximum vertex degree Δ and bounded maximum codegree Δ_2 relative to the average degree. In the divisor hypergraph, Δ ≈ N/2 (the element 1 belongs to ~N^2/2 edges) and Δ_2 is also Θ(N). These parameters make the standard container bound degenerate, yielding containers of size ≈ N and no improvement over the trivial linear bound. Thus the hypergraph container approach cannot produce a sub‑linear upper bound for f(N).
Partition {1,…,N} by residue modulo a carefully chosen integer M (e.g., a product of small primes). Show that if S is large, many of its elements must lie in the same residue class. Within a fixed class, use properties of multiplicative orders modulo M to prove that any two elements have a common divisor structure forcing a violation of the no‑double‑divisor rule unless the class size is bounded. Optimising M gives an explicit lower bound construction (e.g., using numbers with a fixed large prime factor) and improves the known lower bound for the extremal size.
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Model the set A as vertices of a directed graph where an edge i→j exists if i|j and i≠j. The forbidden configuration is a vertex with out‑degree ≥2. This is precisely a collection of stars with centre at each vertex of out‑degree ≤1. By limiting the total number of edges (which is bounded by the sum of floor(n/i)−1) we can apply Turán‑type results for graphs of bounded maximum out‑degree to obtain an upper bound on |A|. The direction will explore known extremal results for bipartite graphs and adapt them to the directed setting, yielding explicit inequalities such as |A| ≤ n / (log₂ n) + O(1).
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Select each integer in [1,n] independently with probability p and then delete elements that cause a violation (i.e., any element that becomes a divisor of two retained numbers). By optimizing p we can show the existence of subsets of size at least c·n / log n for a concrete constant c>0. The work will involve calculating the expected number of deletions using divisor counting functions and applying the Lovász Local Lemma to control dependencies. This yields a constructive lower bound that can be compared with the extremal upper bound.
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Consider the poset ( [1,n], | ) ordered by divisibility. Dilworth’s theorem states that the size of the largest antichain equals the minimum number of chains needed to cover the set. An element that divides two others forces a chain of length at least three. By partitioning [1,n] into the minimum number of chains and analysing how many elements can be kept from each chain while respecting the ‘at most one outgoing edge’ rule, we obtain refined upper bounds. In particular, the approach will examine the structure of chains formed by powers of 2 and other prime powers, leading to bounds of the form |A| ≤ n / (log₂ n) + O(log n).
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