l-trace k-Sperner families of sets

A family of sets F⊆X² is defined to be l-trace k-Sperner if for any subset Y of X with size l the trace of F on Y (the restriction of F to Y) does not contain any chain of length k+1. In this paper we investigate the maximum size that an l-trace k-Sperner family (with underlying set n]={1,2,…,n}) can have for various values of k, l and n.