Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

For each non-negative integer $n$, what antichain(s) in $\{0,1\}^n$ with the pointwise partial order:
$\;\;$ 1. $\;$ have the most elements
$\;\;$ 2. $\;$ minimize the maximum of its elements' sum of coordinates, among those satisfying (1)
?



The obvious candidate is the set of elements whose sum of coordinates is $\: \Big\lfloor \frac{n}2 \Big\rfloor \:$.


Where $S$ is such a set and $\: m = \big\lfloor \log_2(|S|) \big\rfloor \:$, $\;$ I would also want an

efficiently computable injection $\: f : \{0,1\}^m \to S \:$ whose inverse is efficiently computable,

although I imagine that part would be straight-forward.

share|improve this question
add comment

1 Answer

up vote 5 down vote accepted

Your "obvious candidate" is right. this is Sperner's Theorem (not to be confused with Sperner's Lemma).

share|improve this answer
    
The fifth google result for "Sperner's Theorem uniqueness" proves that to be essentially unique, $\hspace{.8 in}$ and the injection is in fact straight-forward to efficiently compute. $\;$ –  Ricky Demer Dec 29 '11 at 22:17
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.