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

Abstract

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 Füredi. We also present lower bounds for some generalisations of the acute angles problem, considering especially some problems concerning colourings of sets of integers.