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Problem 844 — Erdős Problems

Problem 844

Let $A\subseteq \{1,\ldots,N\}$ be such that, for all $a,b\in A$ , the product $ab$ is not squarefree.

Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?

Source: erdosproblems.com/844

Number theory
Intersecting families

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Solved·, Proved

An accepted solution exists. The statement is true.

PROVED (LEAN).

AMS25: B. Alexeev, D. Mixon, and W. Sawin, The independence and clique cover numbers of the squarefree graph. arXiv:2507.01928 (2025).
Ch74: Chvátal, V., Intersecting families of edges in hypergraphs having the hereditary property. (1974), 61-66.
Er92b: Erdős, P., Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) 47 (1992), no. 2, 231--240.

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Status: Solved, Proved
Topics: Number theory
Intersecting families
In the wiki: Wiki page