For positive integers (a,b), write
[
[a,b]=\operatorname{lcm}(a,b)
]
for their least common multiple.

For each integer (N\ge 1), let (f(N)) be the largest size of a set
[
A\subseteq {1,2,\ldots,N}
]
such that there do not exist three distinct elements (a,b,c\in A) satisfying
[
[a,b]=[b,c]=[a,c].
]

Resolve the following Erdős problem completely, at least to the level of its principal explicit question:

Estimate the asymptotic growth of (f(N)). In particular, determine whether
[
f(N)=o(N).
]

Assume for purposes of this task that a complete resolution of the principal question exists, but do not assume in advance that the answer is affirmative or negative. A complete resolution must prove exactly one of the following two statements.

Affirmative resolution:

For every (\varepsilon>0), there exists (N_0(\varepsilon)) such that for every (N\ge N_0(\varepsilon)), every set
[
A\subseteq{1,\ldots,N}
]
with
[
|A|\ge \varepsilon N
]
contains distinct (a,b,c\in A) satisfying
[
[a,b]=[b,c]=[a,c].
]

Equivalently,
[
\lim_{N\to\infty}\frac{f(N)}{N}=0.
]

Negative resolution:

There exists an absolute constant (\delta>0) and infinitely many integers (N) for which there is a set
[
A\subseteq{1,\ldots,N}
]
with
[
|A|\ge \delta N
]
and containing no distinct (a,b,c\in A) satisfying
[
[a,b]=[b,c]=[a,c].
]

Equivalently,
[
\limsup_{N\to\infty}\frac{f(N)}{N}>0.
]

Any unconditional quantitative upper bound of the form
[
f(N)\le \eta(N)N,
\qquad \eta(N)\to 0,
]
is sufficient for an affirmative resolution. Any construction of positive density for infinitely many (N) is sufficient for a negative resolution, even if those (N) form a sparse sequence.

The solution should also obtain the strongest rigorously justified quantitative upper and lower bounds for (f(N)) that follow from its method, and should determine the correct asymptotic order if possible. However, the non-negotiable requirement is to settle the (o(N)) question exactly. Do not claim to have determined the full asymptotic growth unless the proof actually does so.

The forbidden configuration may be reformulated using prime valuations. If
[
a=\prod_p p^{\alpha_p},\qquad
b=\prod_p p^{\beta_p},\qquad
c=\prod_p p^{\gamma_p},
]
where all but finitely many exponents are zero, then
[
[a,b]=[b,c]=[a,c]
]
if and only if, for every prime (p), the maximum of
[
\alpha_p,\beta_p,\gamma_p
]
is attained at least twice.

Equivalently, viewing each positive integer as its finite vector of prime exponents, a forbidden triple is a triple for which no coordinate has a unique strict maximum. This reformulation must be proved carefully before use.

Partial progress does not count unless it implies exactly one of the two resolutions above. In particular, the following are insufficient:

* proving the result only for squarefree integers, smooth integers, rough integers, divisors of a fixed integer, or integers with a bounded number of prime factors;
* proving the result only for sets contained in a restricted interval such as ([N/2,N]);
* proving an affirmative upper bound only for a subsequence of values of (N), without extending it uniformly to all sufficiently large (N);
* proving only that
  [
  f(N)\le (1-\delta)N
  ]
  for some fixed (\delta>0);
* proving an (o(N)) upper bound only under an unproved conjecture;
* constructing sets whose density tends to zero, since this does not disprove (f(N)=o(N));
* constructing one large finite example, since this does not establish positive upper density for infinitely many (N);
* computational verification through any fixed value of (N);
* proving a different problem in which (a,b,c) are allowed to coincide;
* forbidding only triples for which two of the three least common multiples are equal;
* replacing equality of least common multiples by divisibility, approximate equality, or another relaxed relation;
* assuming without proof that extremal sets are squarefree, divisor-closed, primitive, concentrated in a short interval, or supported on a fixed collection of primes;
* reducing the problem to another unproved extremal, density, supersaturation, or container statement of comparable strength.

Standard proved theorems from multiplicative number theory, extremal combinatorics, hypergraph theory, additive combinatorics, sieve theory, probabilistic combinatorics, entropy methods, order theory, or optimization may be used, but they must be stated accurately and applied with all necessary hypotheses and uniformity.

Use multiagent v2 aggressively and dynamically. You have up to 4 concurrent agents available. Do not use a fixed assignment such as “N agents for strategy X.” Instead, manage the search using the following heuristics:

* Begin with a genuinely diverse portfolio of approaches. Agents should explore substantially different formulations, invariants, reductions, prime-exponent vector methods, divisor-lattice formulations, hypergraph independent sets, supersaturation, hypergraph containers, entropy arguments, compression and shifting, smooth-rough decompositions, prime-support methods, density increments, recursive decompositions, random and explicit constructions, CRT mechanisms, linear and integer programming, and computational sanity checks.

* Do not tell most agents the currently favored approach. Preserve independence during early rounds so that agents do not all converge to the same attractive but incomplete divisor-lattice, container, or density argument.

* Maintain an explicit registry of approach families. Group agents by the mathematical idea they are using, not by superficial wording. If many agents converge to one family, redirect some of them toward underexplored formulations.

* Do not allow one approach to dominate merely because it gives an elegant reformulation. A route that ends at an unproved supersaturation, container, removal, or density lemma equivalent in strength to the original problem is not close to completion unless it supplies a genuinely new proof of that lemma.

* When an approach stalls at a theorem-strength missing lemma, mark that route as blocked. Only continue assigning agents to it if someone proposes a materially new mechanism, invariant, decomposition, construction, inequality, or descent argument.

* Keep several incompatible proof routes alive through multiple rounds. Maintain both affirmative-density routes and positive-density construction routes until one side is rigorously ruled out. Cross-pollinate ideas only after independent agents have developed them far enough to expose their real strengths and gaps.

* Use computational agents throughout. They should compute extremal sets for small and medium (N), encode the forbidden triples as a 3-uniform hypergraph, solve exact or relaxed independent-set problems, test conjectured compression principles, search for positive-density constructions, identify extremal patterns, and find counterexamples to proposed intermediate lemmas. Computation is evidence unless it is converted into a rigorous general argument or a finite certificate completing a valid reduction.

* Use adversarial agents throughout. Every candidate proof must be checked for:

  * the requirement that (a,b,c) are distinct;
  * equality of all three pairwise least common multiples;
  * correct use of the prime-valuation reformulation;
  * hidden assumptions of squarefreeness, bounded prime multiplicity, or restricted prime support;
  * loss of uniformity in (N);
  * claiming an affirmative resolution from a bound proved only along a subsequence of (N);
  * rejecting a valid negative resolution merely because its positive-density construction occurs along a sparse infinite sequence;
  * confusing positive density with density tending slowly to zero;
  * using a finite computation as an asymptotic proof;
  * constructions whose elements exceed (N);
  * reductions that do not preserve the interval ({1,\ldots,N});
  * applying a container, supersaturation, removal, or entropy theorem without verifying its hypotheses quantitatively;
  * circular use of a hypergraph-density statement equivalent to the original problem;
  * unproved independence assumptions concerning divisibility by different primes;
  * proving only (f(N)\le (1-\delta)N), which does not imply (f(N)=o(N));
  * claiming a positive-density counterexample from sets whose density actually tends to zero.

* Require agents to return concrete lemmas, constructions, inequalities, equations, algorithms, code outputs, extremal examples, certificates, or counterexamples to proposed sublemmas. Reject status reports, vague optimism, and claims that an unproved global density, supersaturation, or compatibility statement is “routine.”

* The root agent should repeatedly synthesize, challenge, redirect, and launch new rounds. Do not stop after the first wave fails. Produce a complete affirmative proof or a complete positive-density counterexample construction only if it survives adversarial audit; otherwise report only the strongest rigorously proved derivation and its exact remaining gap.

Do not return merely because current approaches fail or agents report theorem-strength gaps. Continue launching new rounds, reopening blocked approaches only when there is a genuinely new mechanism, and searching for fresh formulations.

Return only when the (o(N)) question has been completely resolved and the argument survives adversarial audit. Do not return a reduction, partial result, isolated missing lemma, finite computation, numerical guess, “best effort” summary, or explanation of why the problem is difficult.

Do not stop, return, or give up until you have found a complete and rigorous solution. Continue exploring every plausible approach, repairing failed arguments, and developing new ones until the problem is fully resolved.

Public search may be used only for ordinary mathematical background or standard named theorems, not to search for a solution to this exact Erdős problem or benchmark. Do not search the public web merely to determine whether this problem is open, and do not answer that it is open.