A note on traces of set families

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 $\mathcal{F}{\mathrm{\mid }}_L=\{F\cap L:F\in \mathcal{F}\}$ does not contain any chain of length $k+1$. In this paper we prove that for any positive integers $l^{\mathrm{\prime }},k$ with $l^{\mathrm{\prime }}<k$ if $\mathcal{F}$ is $(n-l^{\mathrm{\prime }})$-trace $k$-Sperner, then $|$\mathcal{F}$| \le (k-l’+o(1))\binom{n}{\lfloor n/2\rfloor}$ and this bound is asymptotically tight.