In the subcritical Schwarzschild-de Sitter (Kottler) spacetimes, each cosmological region is connected to exactly two stationary regions, each connected to a black hole. The same occurs in subcritical Kerr-de Sitter spacetimes. [Hintz 2021] constructed forward solutions to the vacuum Einstein equation containing a cosmological region with multiple Schwarzschild- or Kerr-like black holes, but this construction required the global balance conditions that $\sum_{i=1}^N m_i p_i=0$, where each black hole has limiting mass $m_i$ and position $p_i$ relative to an identification of null infinity with the 3-dimensional unit sphere. The global balance condition is satisfied by the antipodal black holes in the Kerr-de Sitter solution. Since the de Sitter spacetime has particle horizons, it is natural to believe that black hole formation should be local and that black holes should be able to form in arbitrary configurations. This talk will present work on the construction of a solution to the Einstein-scalar field system in spherical symmetry with positive cosmological constant that leads to a single black hole. This involves both gluing of initial data and estimating the (forward) global solution launched by this data. Work in progress on obstructions to the regularity of the solutions and extending this to arbitrarily many black holes with arbitrary configurations will also be discussed.

This work is joint with Pierre De Roubin