  A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated
with each vertices, one can choose a subset of $b$ colors such that adjacent
vertices are colored with disjoint color sets. This paper shows an equivalence
between the $(a,b)$-choosability of a graph and the $(a,b)$-choosability of one
of its subgraphs called the extended core. As an application, this result
allows to prove the $(5,2)$-choosability and $(7,3)$-colorability of
triangle-free induced subgraphs of the triangular lattice.
