  Given a $k$-colouring of the edges of the complete graph $K_n$, are there
$k-1$ monochromatic components that cover its vertices? This important special
case of the well-known Lov\'asz-Ryser conjecture is still open. In this paper
we consider a strengthening of this question, where we insist that the covering
sets are not merely connected but have bounded diameter. In particular, we
prove that for any colouring of $E(K_n)$ with 4 colours, there is a choice of
sets $A_1, A_2, A_3$ that cover all vertices, and colours $c_1, c_2, c_3$, such
that for each $i = 1,2,3$ the monochromatic subgraph induced by the set $A_i$
and the colour $c_i$ has diameter at most 160.
