Two examples of Borel partially ordered sets with the countable chain condition

We define an open symmetric two-place relation on the reals such that the reals cannot be covered by countably many sets of related elements, but there is no uncountable set of unrelated elements. The poset $\mathcal{P}$ of finite sets of related elements satisfies the countable chain condition but it may fail to have the property K, i.e., a substantial irregularity can be injected in $\mathcal{P}$.