A factor of i.i.d with uniform marginals and infinite clusters spanned   by equal labels
  We give an example of an $\mathsf{FIID}$ vertex-labeling of $\mathbb{T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are infinite. 
P\'eter Mester
math.PR