Sets in which the product of any \(k\) elements increased by \(t\) is a \(k\)th-power (Q2719868)

For a fixed integer \(t\), a \(P_t\)-set of size \(n\) is a set \(A= \{x_1,x_2,\dots, x_n\}\) of distinct positive integers such that \(x_ix_j+t\) is a square of an integer whenever \(i\neq j\). Let \(k>1\) be a positive integer; a \(P_t^{(k)}\)-set of size \(n\) is a set \(A= \{x_1,x_2,\dots, x_n\}\) of distinct positive integers such that \(\prod_{i\in I}x_i+t\) is a \(k\)th-power of an integer for each \(I\subset \{1,2,\dots, n\}\) where \(\text{card}(I)= k\). A \(P_t^{(k)}\)-set \(A\) is said to be extendible if there exists an integer \(a\not\in A\) such that \(A\cup \{a\}\) is a \(P_t^{(k)}\)-set. The aims are to: (1) exhibit a \(P_t^{(3)}\)-set of size 4; (2) show this set is nonextendible; (3) prove that the \(P_{-8}^{(4)}\)-set \(\{1,2,3,4\}\) and the \(P_1^{(4)}\)-set \(\{1,2,5,8\}\) are nonextendible; (4) show that any \(P_t^{(k)}\)-set is finite.