Infinite Sperner's theorem
نویسندگان
چکیده
One of the most classical results in extremal set theory is Sperner's theorem, which says that largest antichain Boolean lattice 2[n] has size Θ(2nn). Motivated by an old problem Erdős on growth infinite Sidon sequences, this note we study rate maximum antichains. Using well known Kraft's inequality for prefix codes, it not difficult to show antichains should be “thinner” than corresponding finite ones. More precisely, if F⊂2N antichain, thenliminfn→∞|F∩2[n]|(2nnlogn)−1=0. Our main result shows bound essentially tight, is, construct F such thatliminfn→∞|F∩2[n]|(2nnlogCn)−1>0 holds some absolute constant C>0.
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2022
ISSN: ['0097-3165', '1096-0899']
DOI: https://doi.org/10.1016/j.jcta.2021.105558