Improved bounds on chain saturation

Description of video

Date: 11/30/23
Speaker :Martin Ryan


    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 k6, 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 k7 and the lower bound for k483.

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