Improved bounds on chain saturation

Let sat(n,k) denote the minimum size of a family in
the Boolean lattice of dimension n with the property that there is
no chain with k+1 elements but the addition of one more element to
the family produces such a chain. It is well known that for n
sufficiently large, sat(n,k) is the same value, which we
denote sat(k).

For k≥6, the best known bounds are

due to Gerbner, Keszegh, Lemons, Palmer, P\'alv\"olgyi, and Patk\'os
and to Morrison, Noel and Scott (based on a construction by Gerbner,
et al.), respectively.

We improve both bounds to

The upper bound works for k≥7 and the lower bound for k≥483.

This is joint work with Nick Veldt, Iowa State University.