Sets of points determining only acute angles and some related colouring problems

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