On product representations of squares

Fix k ≥ 2. For any N ≥ 1, let Fₖ(N) denote the cardinality of the largest subset of \1,,N\ that does not contain k distinct elements whose product is a square. Erd{o}s, S\'ark{o}zy, and S\'os showed that F₂(N) = (6/π²+o(1)) N, F₃(N) = (1-o(1))N, Fₖ(N) N/log N for even k ≥ 4, and Fₖ(N) N for odd k ≥ 5. Erd{o}s then asked whether Fₖ(N) = (1-o(1)) N for odd k ≥ 5. Using a probabilistic argument, we answer this question in the negative.