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- W3048768488 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 $Thetabig(frac{2^n}{sqrt{n}}big)$. 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 $mathcal{F}subset 2^{mathbb{N}}$ is an antichain, then $$liminf_{nrightarrow infty}big|mathcal{F} cap 2^{[n]}big|left(frac{2^n}{nlog n}right)^{-1}=0.$$ Our main result shows that this bound is essentially tight, that is, we construct an antichain $mathcal{F}$ such that $$liminf_{nrightarrow infty}big|mathcal{F} cap 2^{[n]}big|left(frac{2^n}{nlog^{C} n}right)^{-1}>0$$ holds for some absolute constant $C>0$." @default.
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- W3048768488 date "2020-08-11" @default.
- W3048768488 modified "2023-09-23" @default.
- W3048768488 title "Infinite Sperner's theorem" @default.
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