A note on traces of set families

Abstract

A family of sets $\mathcal{F} \subseteq 2^{[n]}$ is defined to be $l$-trace $k$-Sperner if for any $l$-subset $L$ of $[n]$ the family of traces $\mathcal{F}|_L=\{F \cap L: F \in \mathcal{F}\}$ does not contain any chain of length $k+1$. In this paper we prove that for any positive integers $l',k$ with $l'<k$ if $\mathcal{F}$ is $(n-l')$-trace $k$-Sperner, then $|\mathcal{F}| \le (k-l'+o(1))\binom{n}{\lfloor n/2\rfloor}$ and this bound is asymptotically tight.