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 $A$ ?
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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.
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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.