A note on infinite antichain density

Abstract

Let $\mathcal{F}$ be an antichain of finite subsets of $\mathbb{N}$. How quickly can the quantities $|\mathcal{F}\cap 2^{[n]}|$ grow as $n\to \infty$? We show that for any sequence $(f_n{)}_{n\ge n_0}$ of positive integers satisfying ${\sum }_{n=n_0}^{\infty }{f}_n/2^n\le 1/4$, $f_{n_0}=1$ and $f_n\le f_{n+1}\le 2f_n$, there exists an infinite antichain $\mathcal{F}$ of finite subsets of $\mathbb{N}$ such that $|\mathcal{F}\cap 2^{[n]}|\ge f_n$ for all $n\ge n_0$. It follows that for any $\epsilon >0$ there exists an antichain $\mathcal{F}\subseteq 2^{\mathbb{N}}$ such that $$\underset{n\to \infty }{\mathrm{liminf}}|\mathcal{F}\cap 2^{[n]}|\cdot {\left(\frac{2^n}{n{\log }^{1+\epsilon }n}\right)}^{-1}>0.$$ This resolves a problem of Sudakov, Tomon and Wagner in a strong form, and is essentially tight.

arXiv commentMinor correction: added condition that $f_{n_0}=1$ to the statement of Theorem 3, 6 pages