Finite Graphs and the Number of Sums and Products

Let G be a graph with k vertices (1, 2, …, k) and e edges. Let A = (α₁,α₂,..,α _k ) be a set of k integers, and let G(A) be the set of all integers of the form α _i + α _j and α _i α _j , where (i,j) is an edge of G. Erdös and Szemerédi conjectured that |G(α)| ≫ _{ε e} /kε for every ε > 0 and every set A. This conjecture will be proved in the case that the diameter of the set A is polynomial in k.