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The Boolean lattice    (   2   [  n  ]    ,  ⊂  )    is the family of all subsets of    [  n  ]  =  {  1  ,  ⋯  ,  n  }   , ordered by inclusion. Let   P   be a partially ordered set. We prove that if   n   is sufficiently large, then there exists a packing   P   of copies of   P   in    (   2   [  n  ]    ,  ⊂  )    that covers almost every element of    2   [  n  ]    :   P   might not cover the minimum and maximum of    2   [  n  ]    , and at most    |  P  |  −  1    additional points due to divisibility. In particular, if    |  P  |    divides2  n   −  2   , then the truncated Boolean lattice2   [  n  ]    −   {  ∅  ,   [  n  ]   }can be partitioned into copies of   P  . This confirms a conjecture of Lonc from 1991.

Packing the Boolean lattice with copies of a poset