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
|
| _version_ | 1797106477168590848 |
|---|---|
| author | Balister, P Powierski, E Scott, A Tan, J |
| author_facet | Balister, P Powierski, E Scott, A Tan, J |
| author_sort | Balister, P |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T07:03:04Z |
| format | Journal article |
| id | oxford-uuid:5285f76f-4e43-4242-af4c-be24ddf707b8 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:03:04Z |
| publishDate | 2022 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:5285f76f-4e43-4242-af4c-be24ddf707b82022-04-07T08:05:59ZA note on infinite antichain densityJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5285f76f-4e43-4242-af4c-be24ddf707b8EnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2022Balister, PPowierski, EScott, ATan, JLet \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. |
| spellingShingle | Balister, P Powierski, E Scott, A Tan, J A note on infinite antichain density |
| title | A note on infinite antichain density |
| title_full | A note on infinite antichain density |
| title_fullStr | A note on infinite antichain density |
| title_full_unstemmed | A note on infinite antichain density |
| title_short | A note on infinite antichain density |
| title_sort | note on infinite antichain density |
| work_keys_str_mv | AT balisterp anoteoninfiniteantichaindensity AT powierskie anoteoninfiniteantichaindensity AT scotta anoteoninfiniteantichaindensity AT tanj anoteoninfiniteantichaindensity AT balisterp noteoninfiniteantichaindensity AT powierskie noteoninfiniteantichaindensity AT scotta noteoninfiniteantichaindensity AT tanj noteoninfiniteantichaindensity |