  We study the following two problems:
  (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set
$A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$?
  (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact
subset $A$ of a Euclidean space have if $A$ does not contain three points that
form an angle in the $\de$-neighborhood of $\al$?
  An interesting phenomenon is that different angles show different behaviour
in the above problems. Apart from the clearly special extreme angles 0 and
$180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special
role in problem (2): the maximal dimension is smaller for these special angles
than for the other angles. In problem (1) the angle $90^\circ$ seems to behave
differently from other angles.
