A note on infinite antichain density
Let \scrF be an antichain of finite subsets of \BbbN . How quickly can the quantities | \scrF \cap 2 [n] | grow as n \rightarrow \infty ? We show that for any sequence (fn)n\geq n0 \sum of positive integers satisfying \infty n=n0 fn/2 n \leq 1/4 and fn \leq fn+1 \leq 2fn, there exists an infinite an...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Society for Industrial and Applied Mathematics
2022
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| Summary: | Let \scrF be an antichain of finite subsets of \BbbN . How quickly can the quantities | \scrF \cap
2
[n]
| grow as n \rightarrow \infty ? We show that for any sequence (fn)n\geq n0 \sum
of positive integers satisfying
\infty
n=n0
fn/2
n \leq 1/4 and fn \leq fn+1 \leq 2fn, there exists an infinite antichain \scrF of finite subsets of
\BbbN such that | \scrF \cap 2
[n]
| \geq fn for all n \geq n0. It follows that for any \varepsilon > 0 there exists an antichain
\scrF \subseteq 2
\BbbN such that lim infn\rightarrow \infty | \scrF \cap 2
[n]
| \cdot \bigl(
2
n
n log1+\varepsilon n
\bigr) - 1 > 0. This resolves a problem of Sudakov,
Tomon, and Wagner in a strong form and is essentially tight. |
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