Sperner families of bounded VC-dimension

Abstract

We explore a problem of Frankl (1989). A family F of subsets of 0–1, 2, …, m is said to have trace Kk if there is a subset S ⊆1, 2, …, m with vbSvb = k so that F ∩ Svb F ∈ F yields all 2k possible subsets. Frankl (1989) conjectured that a family F which is an antichain (in poset given by ⊆ order) and has no trace Kk has maximum size mk−1 for m > 2k − 4 and we verify this for k = 4 by finding the extremal families for k = 2,3. We are attempting to bring together Sperner's theorem and the results of Vapnik and Chervonenkis (1971), Sauer (1972), Perles and Shelah (1972). We also consider the function f(m, k, l), whose value was conjectured by Frankl (1989), which is the maximum size of F with no trace equal to Kk and no chain of size l + 1. We compute f(m, k, k − 1) and for f(m, k, k) provide an unexpected construction for extremal families achieving the bound.