  Colored triangulations offer a generalization of combinatorial maps to higher
dimensions. Just like maps are gluings of polygons, colored triangulations are
built as gluings of special, higher-dimensional building blocks, such as
octahedra, which we call colored building blocks and known in the dual as
bubbles. A colored building block is determined by its boundary triangulation,
which in the case of polygons is simply characterized by its length. In three
dimensions, colored building blocks are labeled by some 2D triangulations and
those homeomorphic to the 3-ball are labeled by the subset of planar ones.
Similarly to Euler's formula in 2D which provides an upper bound to the number
of vertices at fixed number of polygons with given lengths, we look in three
dimensions for an upper bound on the number of edges at fixed number of given
colored building blocks. In this article we solve this problem when all colored
building blocks, except possibly one, are homeomorphic to the 3-ball. To do
this, we find a characterization of the way a colored building block
homeomorphic to the ball has to be glued to other blocks of arbitrary topology
in a colored triangulation which maximizes the number of edges. This local
characterization can be extended to the whole triangulation as long as there is
at most one colored building block which is not a 3-ball. The triangulations
obtained this way are in bijection with trees. The number of edges is given as
an independent sum over the building blocks of such a triangulation. In the
case of all colored building blocks being homeomorphic to the 3-ball, we show
that these triangulations are homeomorphic to the 3-sphere. Those results were
only known for the octahedron and for melonic building blocks before. This
article is self-contained and can be used as an introduction to colored
triangulations and their bubbles from a purely combinatorial point of view.
