Infinite Sperner's theorem

Benny Sudakov,István Tomon,Adam Zsolt Wagner

doi:10.1016/j.jcta.2021.105558

Benny Sudakov, István Tomon + Show 1 more

Open Access

https://doi.org/10.1016/j.jcta.2021.105558

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Abstract

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 Θ(2nn). Motivated by an old problem of Erdős 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⊂2N is an antichain, thenliminfn→∞|F∩2[n]|(2nnlog⁡n)−1=0. Our main result shows that this bound is essentially tight, that is, we construct an antichain F such thatliminfn→∞|F∩2[n]|(2nnlogC⁡n)−1>0 holds for some absolute constant C>0.

Highlights

A typical question in extremal combinatorics asks to determine or estimate the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements
Motivated by an old problem of Erdős on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains
The fact that F is an antichain implies that the family {sA : A ∈ F ∩ 2[N]} is a prefix code

Summary

Introduction

A typical question in extremal combinatorics asks to determine or estimate the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements. Turán [7] that no for all n This was infinite further refined by Erdős who showed (see [8], Chapter II, §3) that if S is a Sidon set, . This shows that infinite Sidon sequences are thinner than the densest finite ones The proof of this result appeared in a 1953 letter from Erdős to Stöhr. A rem√arkable construction of Ruzsa [12] shows that there are Sidon sets S with S ∩ [n] ≥ n 2−1+o(1) for all n Closing this gap is a fascinating open problem. They extended this result to the more general setting of Ks,t-free graphs. In this short note we consider a similar problem for maximal antichains.

Infinite antichains cannot be too large

Constructing dense infinite antichains

Concluding remarks and open problems