Infinite Sperner's theorem

One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice 2([n]) has size Theta(2n/root n) Motivated by an old problem of ErdOs on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains. Using the well known Kraft's inequality for prefix codes, it is not difficult to show that infinite antichains should be "thinner" than the corresponding finite ones. More precisely, if F subset of 2(N) is an antichain, then lim(n -> )(infinity) inf vertical bar F boolean AND 2([n])vertical bar (2(n)/n logCn)(-1) = 0. Our main result shows that this bound is essentially tight, that is, we construct an antichain F such that lim(n -> )(infinity) inf vertical bar F boolean AND 2([n])vertical bar (2(n)/ n log(C) n)(-1) > 0. holds for some absolute constant C > 0. (C) 2021 The Author(s). Published by Elsevier Inc.