# On an extremal problem for poset dimension

## TL;DR

This paper investigates the size of the largest guaranteed d-dimensional subposet within any poset of n elements, improving bounds for the case d=2 and generalizing to higher dimensions.

## Contribution

It provides improved asymptotic bounds for the size of large d-dimensional subposets in arbitrary posets, advancing understanding of poset dimension extremal problems.

## Key findings

- Established that f(n)=O(n^{2/3}) for 2-dimensional subposets.
- Generalized the bound to higher dimensions with f_d(n)=O(n^{d/(d+1)}).
- Improved previous upper bounds for extremal poset dimension problems.

## 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 $f(n)\geqslant n^{1/2}$. 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.