A diameter graph in is a graph whose set of vertices is a finite subset of 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 and on a three-dimensional sphere. We prove an analog of Vazsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than . We prove Schur's conjecture for diameter graphs in We also establish the maximum number of triangles a diameter graph in can have, showing that the extremum is attained only on specific Lenz configurations.