Abstract:   

   Diophantus initiated the study of sets of positive rational numbers with the property that the product of any two of them is one less than the square of a rational number. He found, for example, the set {1/16, 33/16, 17/4, 105/16}.Fermat was apparently the first to find a set of four positive integers with the property that the product of any two of the integers plus one is a square. His example was {1, 3, 8, 120}. In 2004 Dujella showed that there are no sets of six integers and that there are only finitely many sets of five positive integers with the above property. Let V denote the set of squares, cubes and higher powers of positive integers. In 2002 Gyarmati, Sarkozy and Stewart asked how large a set of positive integers A can be if ab + 1 is in V whenever a and b are distinct elements of A. They conjectured that there is an absolute bound for the cardinality of A and Luca has shown that this is a consequence of the abc conjecture. In this talk we shall discuss recent progress towards the conjecture. The best results to date have been obtained by a combination of methods from Diophantine approximation and extremal graph theory.