On the sizes of $t$-intersecting $k$-chain-free families

A set system $\mathcal{F}$ is $t$-$\mathit{\text{intersecting}}$, if the size of the intersection of every pair of its elements has size at least $t$. A set system $\mathcal{F}$ is $k$-$\mathit{\text{Sperner}}$, if it does not contain a chain of length $k+1$. Our main result is the following: Suppose that $k$ and $t$ are fixed positive integers, where $n+t$ is even with $t\le n$ and $n$ is large enough. If $\mathcal{F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sperner family, then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }$ has size at most the size of the sum of $k$ layers, of sizes $(n+t)\mathrm{/}2,\dots ,(n+t)\mathrm{/}2+k-1$. This bound is best possible. The case when $n+t$ is odd remains open.