Largest family without a pair of posets on consecutive levels of the Boolean lattice

Abstract

Suppose $k\ge 2$ is an integer. Let $Y_k$ be the poset with elements $x_1,x_2,y_1,y_2,\dots ,y_{k-1}$ such that $y_1<y_2<\cdots <y_{k-1}<x_1,x_2$ and let $Y_k^{\prime }$ be the same poset but all relations reversed. We say that a family of subsets of $[n]$ contains a copy of $Y_k$ on consecutive levels if it contains $k+1$ subsets $F_1,F_2,G_1,G_2,\dots ,G_{k-1}$ such that $G_1\subset G_2\subset \cdots \subset G_{k-1}\subset F_1,F_2$ and $|F_1|=|F_2|=|G_{k-1}|+1=|G_{k-2}|+2=\cdots =|G_1|+k-1$. If both $Y_k$ and $Y_k^{\prime }$ on consecutive levels are forbidden, the size of the largest such family is denoted by ${\mathrm{La}}_{\mathrm{c}}(n,Y_k,Y_k^{\prime })$. In this paper, we will determine the exact value of ${\mathrm{La}}_{\mathrm{c}}(n,Y_k,Y_k^{\prime })$.