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 | ≤ 2n+ 1 is n, provided n ≥ 2, and this inequality is tight when P contains the standard example Sn. However, there are posets with large dimension that do not contain the standard example S2. Moreover, for each fixed d ≥ 2, if P is a poset with |P | ≤ 2n+1 and P does not contain the standard example Sd, then dim(P ) = o(n). Also, for large n, there is a poset P with |P | = 2n and dim(P ) ≥ (1 − o(1))n such that the largest d so that P contains the standard example Sd is o(n). In this paper, we will show that for every integer c ≥ 1, there is an integer f(c) = O(c2) so that for large enough n, if P is a poset with |P | ≤ 2n+ 1 and dim(P ) ≥ n− c, then P contains a standard example Sd with d ≥ n − f(c). From below, we show that f(c) = Ω(c4/3). 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.