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

A set system F is t−intersecting, if the size of the intersection of every pair of its elements has size at least t.
A set system F is k−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≤n and n is large enough.
If F⊆2[n] is a t-intersecting k-Sperner family, then |F| has size at most  the size of the sum of
k layers, of sizes (n+t)/2,…,(n+t)/2+k−1.

This bound is best possible. The case when n+t is odd remains open.

Joint work with Józsi Balogh and Will Linz.