A Note on Infinite Antichain Density

Let F be an antichain of finite subsets of N. How quickly can the quantities vertical bar F boolean AND 2([n])vertical bar grow as n -> infinity? We show that for any sequence (f(n))(n >= n0) of positive integers satisfying Sigma(infinity)(n=n0)f(n)/2(n) <= 1/4 and f(n) <= f(n+1) <= 2f(n), there exists an infinite antichain F of finite subsets of N such that vertical bar F boolean AND 2([n])vertical bar >= f(n) for all n >= n(0). It follows that for any epsilon > 0 there exists an antichain F subset of 2(N) such that lim inf(n ->infinity) vertical bar F boolean AND 2([n])vertical bar . (2(n)/n log1+epsilon n)(-1) > 0. This resolves a problem of Sudakov, Tomon, and Wagner in a strong form and is essentially tight.