The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of (n+1)-element chains in the power set ℘({1,2,…,n}). We will show that for each fixed α>0 there is a family of α2n sets containing (α+o(1))n! such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.
Paper (pdf)