Generalizations and Strengthenings of Ryser's Conjecture

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem only known to be true $r\leq 3$, or when $\nu(H)=1$ and 5$. An equivalent formulation in $r$-edge coloring a graph $G$ independence $\alpha(G)$, there exists $(r-1)\alpha(G)$ monochromatic connected subgraphs which set $G$. 
 We make case this latter naturally leads variety stronger conjectures generalizations hypergraphs multipartite graphs. Regarding these strengthenings, we survey results, improving upon some, introduce collection new problems results.