The Lubell-Yamamoto-Meshalkin inequality, commonly abbreviated the LYM inequality, is a bound for an antichain 
of subsets of an 
-element set 
. It states that
 |
For a short proof, choose a maximal chain of subsets of 
uniformly at random. A fixed 
-element set belongs to the chain
with probability 
. Since a chain meets an antichain in at most one member, summing these probabilities
over 
gives the inequality.
Since 
for every 
, the LYM inequality implies Sperner's
theorem.
