Let $g(n)$ denote the least integer such that among any $g(n)$ points in
general position in the plane there are always $n$ in convex position.
In 1935 P. Erd\H os and G. Szekeres showed that $g(n)$ exists and
$2^{n-2}+1\le g(n)\le {2n-4\choose n-2}+1$.
Recently, the upper bound has been slightly improved by Chung and
Graham and by Kleitman and Pachter.
In this note we further improve the upper bound to
$$g(n)\le {2n-5\choose n-2}+2.$$
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