Improved bounds in Weaver and Feichtinger Conjectures

We sharpen the constant in the $K{S}_2$ conjecture of Weaver \cite{We}, which was validated by Marcus, Spielman, and Srivastava \cite{MSS} in their solution of the Kadison–Singer problem. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture.