The spanning-coloring covering conjecture for complete partite hypergraphs

Let $r\geq 3$ and $t\geq 1$. A complete $r$-uniform $r$-partite hypergraph has its vertices partitioned into $r$ nonempty classes, with edges consisting of one vertex from each class. An edge-coloring is spanning if every vertex is incident to an edge of every color used. A monochromatic component is a connected component of the subhypergraph formed by the edges of one color, and let $\mathrm{cov}(r,k)$ be the maximum, over spanning $k$-colorings, of the minimum number of monochromatic components needed to cover the vertex set.

The complete-partite covering conjecture.

$$\mathrm{cov}(r,r+t)=t+1$$

for every $r\geq 3$ and $t\geq 1$.

The conjecture is proved for $1\leq t\leq r-1$, while for $t\geq r$ the paper proves only that $t+2$ monochromatic components suffice; consequently the case corresponding to $k\geq 2r$ remains open.