 
Summary: SUMS AND PRODUCTS ALONG SPARSE GRAPHS
NOGA ALON, OMER ANGEL, ITAI BENJAMINI, AND EYAL LUBETZKY
Abstract. In their seminal paper from 1983, Erdos and Szemer´edi
showed that any n distinct integers induce either n1+
distinct sums of
pairs or that many distinct products, and conjectured a lower bound
of n2o(1)
. They further proposed a generalization of this problem, in
which the sums and products are taken along the edges of a given graph
G on n labeled vertices. They conjectured a version of the sumproduct
theorem for general graphs that have at least n1+
edges.
In this work, we consider sumproduct theorems for sparse graphs,
and show that this problem has important consequences already when
G is a matching (i.e., n/2 disjoint edges): Any lower bound of the form
n1/2+
for its sumproduct over the integers implies a lower bound of
n1+
for the original ErdosSzemer´edi problem.
In contrast, over the reals the minimal sumproduct for the match
