On large F-Diophantine sets

Let F∈ Z[x, y] and m≥ 2 be an integer. A set A⊂ Z is called an (F, m)-Diophantine set if F(a, b) is a perfect m-power for any a, b∈ A where a≠ b. If F is a bivariate polynomial for which there exist infinite (F, m)-Diophantine sets, then there is a complete qualitative characterization of all such polynomials F. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers S of size n, there are infinitely many bivariate polynomials F such that S is an (F, 2)-Diophantine set. In addition, we show that the degree of F can be as small as 4 ⌊ n/ 3 ⌋.