
> 
> 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.
  
  




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


