Hadwiger's conjecture for graphs

Let $G$ be a finite graph and let $t\in\mathbb{N}$. A minor of $G$ is a graph obtainable from $G$ by a sequence of vertex deletions, edge deletions, and edge contractions; write $\chi(G)$ for the chromatic number and $K_t$ for the complete graph on $t$ vertices. Hadwiger's conjecture. If $\chi(G)>t$, then $G$ has $K_t$ as a minor. Equivalently, if $G$ does not have $K_t$ as a minor, then $\chi(G)<t$. This is a central open problem in graph theory, formulated by Hadwiger in 1943; the source presents it as an open conjecture, while the paper studies its analogue for hypergraphs.