In their seminal paper Erdos and Szemeredi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this talk we show that this strong form of the Erdos and Szemeredi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets and ratio sets along the edges of graphs.
Joint work with Noga Alon and Imre Ruzsa.