2019/08/03 by Martin, Ryan R., Smith, Heather C., Walker, Shanise · 1 citation
#05D05 #06A07 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.01108
Given a finite poset P, a family F of elements in the Boolean lattice is induced-P-saturated if F contains no copy of P as an induced subposet but every proper superset of F contains a copy of P as an induced subposet. The minimum size of an induced-P-saturated family in the n-dimensional Boolean lattice, denoted sat^*(n,P), was first studied by Ferrara et al. (2017). Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove sat^*(n,D2)≥√(n), improving upon a logarithmic lower bound. For the antichain with k+1 elements, we prove sat^*(n,Ak+1)≥ (1-ok(1))(kn)/(log2 k), improving upon a lower bound of 3n-1 for k≥ 3.