  The problem of constructing dense subsets S of {1,2,..,n} that contain no
arithmetic triple was introduced by Erdos and Turan in 1936. They have
presented a construction with |S| = \Omega(n^{\log_3 2}) elements. Their
construction was improved by Salem and Spencer, and further improved by Behrend
in 1946. The lower bound of Behrend is |S| = Omega({n \over {2^{2 \sqrt{2}
\sqrt{\log_2 n}} \cdot \log^{1/4} n}}). Since then the problem became one of
the most central, most fundamental, and most intensively studied problems in
additive number theory. Nevertheless, no improvement of the lower bound of
Behrend was reported since 1946.
  In this paper we present a construction that improves the result of Behrend
by a factor of Theta(\sqrt{\log n}), and shows that |S| = Omega({n \over {2^{2
\sqrt{2} \sqrt{\log_2 n}}}} \cdot \log^{1/4} n). In particular, our result
implies that the construction of Behrend is not optimal.
  Our construction and proof are elementary and self-contained.
