Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset $P$, denoted $\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\le 2n+1$ is $n$, provided $n\ge2$, and this inequality is tight when $P$ contains the standard example $S_n$. However, there are posets with large dimension that do not contain the standard example $S_2$. Moreover, for each fixed $d\ge2$, if $P$ is a poset with $|P|\le 2n+1$ and $P$ does not contain the standard example $S_d$, then $\dim(P)=o(n)$. Also, for large $n$, there is a poset $P$ with $|P|=2n$ and $\dim(P)\ge (1-o(1))n$ such that the largest $d$ so that $P$ contains the standard example $S_d$ is $o(n)$. In this paper, we will show that for every integer $c\ge1$, there is an integer $f(c)=O(c^2)$ so that for large enough $n$, if $P$ is a poset with $|P|\le 2n+1$ and $\dim(P)\ge n-c$, then $P$ contains a standard example $S_d$ with $d\ge n-f(c)$. From below, we show that $f(c)=\Omega(c^{4/3})$. On the other hand, we also prove an analogous result for fractional dimension, and in this setting $f(c)$ is linear in $c$. Here the result is best possible up to the value of the multiplicative constant.