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

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?

A problem of Erdős and Sárközy.

Weisenberg has provided the following positive proof. It is clear that such a maximal $A$ must contain all non-squarefree numbers. It therefore suffices to find the largest size of a subset of all squarefree numbers in $\{1,\ldots,N\}$ such that any two have at least one prime factor in common. By the result of Chvátal [Ch74] discussed in [701] this is maximised by the set of all even squarefree numbers.

An alternative proof was independently found by Alexeev, Mixon, and Sawin [AMS25].