On an Extremal Problem for Poset Dimension

Grzegorz Guspiel, Piotr Micek, Adam Polak

Abstract: Let f(n) be the largest integer such that every poset on n elements has a 2-dimensional subposet on f(n) elements. What is the asymptotics of f(n)? It is easy to see that f(n) = n 1/2. We improve the best known upper bound and show f(n) = O (n 2/3). For higher dimensions, we show f d ( n ) = O n d d + 1 $f_{d}(n)=\O \left (n^{\frac {d}{d + 1}}\right )$ , where f d (n) is the largest integer such that every poset on n elements has a d-dimensional subposet on f d (n) elements.