Clustering in the three and four color cyclic particle systems in one dimension

We study the $kappa$-color cyclic particle system on the one-dimensional integer lattice $mathbb{Z}$, first introduced by Bramson and Griffeath in cite{bramson1989flux}. In that paper they show that almost surely, every site changes its color infinitely often if $kappain {3,4}$ and only finitely many times if $kappage 5$. In addition, they conjecture that for $kappain {3,4}$ the system clusters, that is, for any pair of sites $x,y$, with probability tending to 1 as $ttoinfty$, $x$ and $y$ have the same color at time $t$. Here we prove that conjecture.