On large $F$-Diophantine sets

Abstract

Let $F\in \mathbb{Z}[x,y]$ and $m\ge 2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. 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 $4\lfloor n/3\rfloor$.