Disproof of a Conjecture by Woodall

In 2001, Woodall conjectured that for every pair of integers $s,t\ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable. In this note we refute this conjecture in a strong form: We prove that for every choice of constants $\epsilon >0$ and $C\ge 1$ there exists $N=N(\varepsilon,C) \in $\mathbb{N}$$suchthatforallintegers$s,t $with$N \le s \le t \le Cs$ there exists a graph without a $K_{s,t}$-minor and list chromatic number greater than $(1-\epsilon )(2s+t)$.