Leírás
Given a finite poset $P$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$P$-saturated if $\mathcal{F}$ contains no copy of $P$ as an induced subposet but every proper superset of $\mathcal{F}$ contains a copy of $P$ as an induced subposet. The minimum size of an induced-$P$-saturated family in the $n$-dimensional Boolean lattice is denoted ${\rm sat}^∗ (n,P)$. Upper bounds follow from constructions and our work focuses on strengthening lower bounds.
For the 4-point poset known as the diamond, we prove ${\rm sat}^∗ (n, \Diamond) \geq\sqrt{n}$, improving upon a logarithmic lower bound that applies for a large class of posets $P$. This is, however, still far from the upper bound of $n+1$.
For the antichain with $k+1$ elements, we prove ${\rm sat}^∗ (n, A_{k+1}) \geq (1−o_k(1)) k n/log_2 k$ , improving upon a lower bound of $3n – 1$ for $k \geq 3$. This contrasts with an upper bound of $(n-1)k - (1/2)\log_2 k - (1/2)\log_2\log_2 k - O(1)$.
This talk includes joint work with Michael Ferrara, Bill Kay, Lucas Kramer, Benjamin Reiniger, Heather C. Smith, Eric Sullivan, and Shanise Walker