l-trace k-Sperner families of sets

A family of sets F ⊆ 2 X 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.