Erdős Problem #121

Let \(F_{k}(N)\) be the size of the largest \(A\subseteq \{1,\ldots,N\}\) such that the product of no \(k\) many distinct elements of \(A\) is a square. Is \(F_5(N)=(1-o(1))N\)? Conjectured by Erdős, Sós, and Sárközy [ESS95], who proved \(F_2(N)=\left(\frac{6}{\pi^2}+o(1)\right)N\), \(F_3(N) = (1-o(1))N\), and also established asymptotics for \(F_k(N)\) for all even \(k\geq 4\) (in particular \(F_k(N)\asymp N/\log N\) for all even \(k\geq 4\)). Erdős [Er38] earlier proved that \(F_4(N)=o(N)\) - indeed, if \(\lvert A\rvert \gg N\) and \(A\subseteq \{1,\ldots,N\}\) then there is a non-trivial solution to \(ab=cd\) with \(a,b,c,d\in A\). This problem was answered in the negative by Tao [Ta24], who proved that for any \(k\geq 4\) there is some constant \(c_k>0\) such that \(F_k(N) \leq (1-c_k+o(1))N\). See also [888](https://www.erdosproblems.com/888).