On an Extremal Problem for Poset Dimension

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) = n1/2. We improve the best known upper bound and show f(n) = O (n2/3). For higher dimensions, we show fd(n)=Ondd+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{d}(n)=\O \left (n^{\frac {d}{d + 1}}\right )$\end{document}, where fd(n) is the largest integer such that every poset on n elements has a d-dimensional subposet on fd(n) elements.