  We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term
arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some
absolute constant $c>0$. In particular, this proves the first non-trivial case
of a conjecture of Erd\H{o}s on arithmetic progressions.
