  In this paper we establish the global in time existence and uniqueness of
solutions to the Boltzmann hierarchy, a hierarchy of equations instrumental for
the rigorous derivation of the Boltzmann equation from many particles. Inspired
by available $L^{\infty}$-based a-priori estimate for solutions to the
Boltzmann equation, we develop the polynomially weighted $L^\infty$ a-priori
bounds for solutions to the Boltzmann hierarchy and handle the factorial growth
of the number of terms in the Dyson's series by reorganizing the sum through a
combinatorial technique known as the Klainerman-Machedon board game argument.
This paper is the first work that exploits such a combinatorial technique in
conjunction with an $L^{\infty}$-based estimate to prove uniqueness of the mild
solutions to the Boltzmann hierarchy. Our proof of existence of global in time
mild solutions to the Boltzmann hierarchy for admissible initial data is
constructive and it employs known global in time solutions to the Boltzmann
equation via a Hewitt-Savage type theorem.
