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We present both probabilistic and constructive lower bounds on the maximum size of a set of points ${\cal S} \subseteq {\Bbb R}^d$ such that every angle determined by three points in ${\cal S}$ is acute, considering especially the case ${\cal S} \subseteq\{0,1\}^d$. These results improve upon a probabilistic lower bound of Erdős and Furedi. We also present lower bounds for some generalisations of the acute angles problem, considering especially some problems concerning colourings of sets of integers.

Sets of Points Determining Only Acute Angles and Some Related Colouring Problems