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SUMS AND PRODUCTS ALONG SPARSE GRAPHS NOGA ALON, OMER ANGEL, ITAI BENJAMINI, AND EYAL LUBETZKY
 

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 n2-o(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 sum-product
theorem for general graphs that have at least n1+
edges.
In this work, we consider sum-product 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 sum-product over the integers implies a lower bound of
n1+
for the original Erdos-Szemer´edi problem.
In contrast, over the reals the minimal sum-product for the match-

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics