Source: Duke Math. J. Volume 129, Number 1
(2005), 129-155.
For two finite sets of real numbers $A$ and $B$, one says that $B$
is sum-free with respect to $A$ if the sum set $\{b+b'\mid b, b'\in B, b\neq b'\}$ is disjoint from $A$. Forty years ago, Erdőos and Moser posed the following
question. Let $A$ be a set of $n$
real numbers. What is the size of the largest subset $B$ of $A$
which is sum-free with respect to $A$?
In this paper, we show that any set $A$ of $n$ real numbers
contains a set $B$ of cardinality at least $g(n) \ln n $ which is
sum-free with respect to $A$, where $g(n)$ tends to infinity with
$n$. This improves earlier bounds of Klarner, Choi, and Ruzsa and
is the first superlogarithmic bound for this problem.
Our proof combines tools from graph theory together with several
fundamental results in additive number theory such as Freiman's
inverse theorem, the Balog-Szemerédi theorem, and Szemerédi's
result on long arithmetic progressions. In fact, in order to
obtain an explicit bound on $g(n)$, we use the recent versions of
these results, obtained by Chang and by Gowers, where significant
quantitative improvements have been achieved.
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