Summary: We show that the size of sets \(\mathcal A\) having the property that with some non-zero integer \(n\), \(a_1a_2+n\) is a perfect power for any distinct \(a_1,a_2\in\mathcal A\), cannot be bounded by an absolute constant. We give a much more precise statement as well, showing that such a set \(\mathcal A\) can be relatively large. We further prove that under the \(abc\)-conjecture a bound for the size of \(\mathcal A\) depending on \(n\) can already be given. Extending a result of Y. Bugeaud and A. Dujella [Math.Proc.Camb.Philos.Soc. 135, No. 1, 1–10 (2003; Zbl 1042.11019)], we also derive an explicit upper bound for the size of \(\mathcal A\) when the shifted products \(a_1a_2+n\) are \(k\)-th powers with some fixed \(k\geq 2\). The latter result plays an important role in some of our proofs, too.