A Note on Covering Edge Colored Hypergraphs by Monochromatic Components

For r ≥ 2, α ≥ r − 1 and k ≥ 1, let c(r, α, k) be the smallest integer c such that the vertex set of any non-trivial r-uniform k-edge-colored hypergraph H with α(H) = α can be covered by c monochromatic connected components. Here α(H) is the maximum cardinality of a subset A of vertices in H such that A does not contain any edges. An old conjecture of Ryser is equivalent to c(2, α, k) = α(r − 1) and a recent result of Z. Király states that c(r, r − 1, k) = dkr e for any r ≥ 3. Here we make the first step to treat non-complete hypergraphs, showing that c(r, r, r) = 2 for r ≥ 2 and c(r, r, r + 1) = 3 for r ≥ 3.