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The claim. Let $f(N)$ be the largest size of a subset of $\{1,\ldots,N\}$ with no three distinct elements $a,b,c$ such that $[a,b]=[a,c]=[b,c]$, the function of Problem 536. Then $f(N)\le(43/48+o(1))N$. This is Theorem 1 of B. Saturnino, An improved elementary constant-density bound for an lcm-triangle problem of Erdős (dated 3 May 2026, hosted on a file-sharing page linked from the site's thread). The proof packs $5N/48+o(N)$ pairwise disjoint lcm triangles from the three templates $\{2m,3m,6m\}$, $\{4m,5m,20m\}$ and $\{7m,9m,63m\}$, separated by conditions on the $2$-, $3$-, $5$- and $7$-adic valuations of $m$; a triangle-free set omits one element of each. The note adds that no argument using only pairwise disjoint forbidden triples can give an upper bound below $2N/3$.

Covers. The upper-bound constant $43/48$ only; neither $f(N)=o(N)$ nor the order of $f(N)$ is settled.

Claimant and postings. The thread post of 3 May 2026 that announced the bound, by the account InfiniteInsights, links the note as the full paper. A reader replied the same day that a standard check found no issues. The site's commentary, last edited 29 April 2026, does not record the bound, and no refereed version was found.

Depends on. No page of this wiki.