It is also of interest to study the extremal numbers of triangular cells in arrangements that may not necessarily be simplicial. In any arrangement, there must be at least n triangles; every arrangement that has only n triangles must be simple.; ; . For Euclidean rather than projective arrangements, the minimum number of triangles is n-2, by Roberts's triangle theorem. The maximum possible number of triangular faces in a simple arrangement is known to be upper bounded by n(n-1)/3 and lower bounded by n(n-3)/3; the lower bound is achieved by certain subsets of the diagonals of a regular 2n-gon.; . For non-simple arrangements the maximum number of triangles is similar but more tightly bounded.; ; . The closely related Kobon triangle problem asks for the maximum number of non-overlapping finite triangles in an arrangement in the Euclidean plane, not counting the unbounded faces that might form triangles in the projective plane. For some but not all values of n, n(n-2)/3 triangles are possible.
