Shifted products that are coprime pure powers

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab+1 of two different elements a and b of A is always a pure power, and the occurring pure powers are all coprime. We prove that each coprime Diophantine powerset A⊂{1,…,N} has |A|⩽8000logN/loglogN for sufficiently large N. The proof combines results from extremal graph theory with number theory. Assuming the famous abc-conjecture, we are able to both drop the coprimality condition and reduce the upper bound to cloglogN for a fixed constant c.