Three Layer Q 2-Free Families in the Boolean Lattice

We prove that the largest Q ₂-free family of subsets of [n] which contains sets of at most three different sizes has at most $big(3 + 2sqrt {3} big)N/3 + o(N) approx 2.1547N,, + o(N)$ members, where $N = { n choose {{lfloor} n/2 {rfloor} }}$ . This improves an earlier bound of 2.207N?+?o(N) by Axenovich, Manske, and Martin.