On an extremal problem for poset dimension

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 $f(n)=\mathcal{O}(n^{2/3})$. For higher dimensions, we show $f_d(n)=\mathcal{O}\left(n^{\frac{d}{d+1}}\right)$, 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.

removed proof of Theorem 3 duplicating previous work; fixed typos and references