On product representations of squares

Fix \(k \geq 2\). For any \(N \geq 1\), let \(F_k(N)\) denote the cardinality of the largest subset of \(\{1,\dots,N\}\) that does not contain \(k\) distinct elements whose product is a square. Erdős, Sárközy, and Sós showed that \(F_2(N) = (\frac{6}{\pi^2}+o(1)) N\), \(F_3(N) = (1-o(1))N\), \(F_k(N) \asymp N/\log N\) for even \(k \geq 4\), and \(F_k(N) \asymp N\) for odd \(k \geq 5\). Erdős then asked whether \(F_k(N) = (1-o(1)) N\) for odd \(k \geq 5\). Using a probabilistic argument, we answer this question in the negative.