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Colored triangulations offer a generalization of combinatorial maps to higherdimensions. Just like maps are gluings of polygons, colored triangulations arebuilt as gluings of special, higher-dimensional building blocks, such asoctahedra, which we call colored building blocks and known in the dual asbubbles. A colored building block is determined by its boundary triangulation,which in the case of polygons is simply characterized by its length. In threedimensions, colored building blocks are labeled by some 2D triangulations andthose 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 numberof vertices at fixed number of polygons with given lengths, we look in threedimensions for an upper bound on the number of edges at fixed number of givencolored building blocks. In this article we solve this problem when all coloredbuilding blocks, except possibly one, are homeomorphic to the 3-ball. To dothis, we find a characterization of the way a colored building blockhomeomorphic to the ball has to be glued to other blocks of arbitrary topologyin a colored triangulation which maximizes the number of edges. This localcharacterization can be extended to the whole triangulation as long as there isat most one colored building block which is not a 3-ball. The triangulationsobtained this way are in bijection with trees. The number of edges is given asan independent sum over the building blocks of such a triangulation. In thecase of all colored building blocks being homeomorphic to the 3-ball, we showthat these triangulations are homeomorphic to the 3-sphere. Those results wereonly known for the octahedron and for melonic building blocks before. Thisarticle is self-contained and can be used as an introduction to coloredtriangulations and their bubbles from a purely combinatorial point of view.

Maximizing the number of edges in three-dimensional colored triangulations whose building blocks are balls