Forcing Posets with Large Dimension to Contain Large Standard Examples

Abstract

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 $n\ge 2$, 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 $d\ge 2$, 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 $c\ge 1$, 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 $f(c)={\varOmega }(c^{4/3})$. 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.