  The general position number of a graph $G$ is the size of the largest set of
vertices $S$ such that no geodesic of $G$ contains more than two elements of
$S$. The monophonic position number of a graph is defined similarly, but with
`induced path' in place of `geodesic'. In this paper we investigate some
extremal problems for these parameters. Firstly we discuss the problem of the
smallest possible order of a graph with given general and monophonic position
numbers. We then determine the asymptotic order of the largest size of a graph
with given general or monophonic position number, classifying the extremal
graphs with monophonic position number two. Finally we establish the possible
diameters of graphs with given order and monophonic position number.
