  A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices
is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs
of vertices that are at diameter apart. This paper is devoted to the study of
different extremal properties of diameter graphs in $\mathbb R^4$ and on a
three-dimensional sphere. We prove an analogue of V\'azsonyi's and Borsuk's
conjecture for diameter graphs on a three-dimensional sphere with radius
greater than $1/\sqrt 2$. We prove Schur's conjecture for diameter graphs in
$\mathbb R^4.$ We also establish the maximum number of triangles a diameter
graph in $\mathbb R^4$ can have, showing that the extremum is attained only on
specific Lenz configurations.
