Some time ago I got an answer to this old exam question addressing cardinals. Now I found the similar question.
Let $\kappa$ be a measurable cardinal, and $\preceq$ a partial ordering on $\kappa$. Prove that first and/or second property holds.
- exists an $A \subset \kappa$ of cardinality $\kappa$ s.t. $A$ is an antichain which is a set of pairwise incomparable elements.
- there exists $L \subset \kappa$ of cardinality $\kappa$ s.t. $\preceq$ is linear on $L$.
This question can be solved using a similar technique to the one illustrated in the previous question. Can we say that since $\kappa$ is measurable, it is Ramsey, and therefore we have an homogeneous set of cardinality $\kappa$ for the function: $$ f(\{x,y\}) = \begin{cases} 1 & (x \preceq y) \vee (y \preceq x)\\ 0 & \text{else} \end{cases} $$ and be done with it?