# Forbidding intersection patterns between layers of the cube

Research output: Contribution to journal › Article › peer-review

## Standard

**Forbidding intersection patterns between layers of the cube.** / Long, Eoin.

Research output: Contribution to journal › Article › peer-review

## Harvard

*Journal of Combinatorial Theory, Series A*, vol. 134, pp. 103-120. https://doi.org/10.1016/j.jcta.2014.08.008

## APA

*Journal of Combinatorial Theory, Series A*,

*134*, 103-120. https://doi.org/10.1016/j.jcta.2014.08.008

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## Author

## Bibtex

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## RIS

TY - JOUR

T1 - Forbidding intersection patterns between layers of the cube

AU - Long, Eoin

PY - 2015/8/1

Y1 - 2015/8/1

N2 - A family ${\cal A}$ is said to be an antichain if $A \not \subset B$ for all distinct $A, B \in {\cal A}$. A classic result of Sperner shows that such families satisfy $|{\cal A}| \leq \binom {n}{\lfloor n/2\rfloor }$, which is easily seen to be best possible. One can view the antichain condition as a restriction on the intersection sizes between sets in different layers of ${\cal P}[n]$. More generally one can ask, given a collection of intersection restrictions between the layers, how large can families respecting these restrictions be? Answering a question of Kalai [8], we show that for most collections of such restrictions, layered families are asymptotically largest. This extends results of Leader and the author from [11].

AB - A family ${\cal A}$ is said to be an antichain if $A \not \subset B$ for all distinct $A, B \in {\cal A}$. A classic result of Sperner shows that such families satisfy $|{\cal A}| \leq \binom {n}{\lfloor n/2\rfloor }$, which is easily seen to be best possible. One can view the antichain condition as a restriction on the intersection sizes between sets in different layers of ${\cal P}[n]$. More generally one can ask, given a collection of intersection restrictions between the layers, how large can families respecting these restrictions be? Answering a question of Kalai [8], we show that for most collections of such restrictions, layered families are asymptotically largest. This extends results of Leader and the author from [11].

U2 - 10.1016/j.jcta.2014.08.008

DO - 10.1016/j.jcta.2014.08.008

M3 - Article

VL - 134

SP - 103

EP - 120

JO - Journal of Combinatorial Theory, Series A

JF - Journal of Combinatorial Theory, Series A

SN - 0097-3165

ER -