Saturation of k-chains in the Boolean lattice

Given a set X, a collection ℱ⊂𝒫(X) is said to be k-Sperner if it does not contain a chain of length k+1 under set inclusion and it is saturated if it is maximal with respect to this probability. Gerbner et al. proved that the smallest saturated k-Sperner system contains at least 2^k/2-1 elements, and later, Morrison, Noel, and Scott showed that the smallest such set contains no more than 2^0.976723k elements. We improve both the upper and lower bounds, showing that the size of the smallest saturated k-Sperner system lies between √(k)2^k/2 and 2^0.950978k.