Largest Family Without a Pair of Posets on Consecutive Levels of the Boolean Lattice

Abstract

AbstractSuppose k ≥ 2 is an integer. Let Yk be the poset with elements x1,x2,y1,y2,…,yk− 1 such that y1 < y2 < ⋯ < yk− 1 < x1,x2 and let $Y_{k}^{\prime }$ Y k ′ be the same poset but all relations reversed. We say that a family of subsets of [n] contains a copy of Yk on consecutive levels if it contains k + 1 subsets F1,F2,G1,G2,…,Gk− 1 such that G1 ⊂ G2 ⊂⋯ ⊂ Gk− 1 ⊂ F1,F2 and |F1| = |F2| = |Gk− 1| + 1 = |Gk− 2| + 2 = ⋯ = |G1| + k − 1. If both Yk and $Y^{\prime }_{k}$ Y k ′ on consecutive levels are forbidden, the size of the largest such family is denoted by $\text {La}_{\mathrm {c}}\left (n, Y_{k}, Y^{\prime }_{k}\right )$ La c n , Y k , Y k ′ . In this paper, we will determine the exact value of $\text {La}_{\mathrm {c}}\left (n, Y_{k}, Y^{\prime }_{k}\right )$ La c n , Y k , Y k ′ .