Erdős Problem #632

A graph is $(a,b)$-choosable if for any assignment of a list of $a$ colours to each of its vertices there is a subset of $b$ colours from each list such that the subsets of adjacent vertices are disjoint.

If $G$ is $(a,b)$-choosable then $G$ is $(am,bm)$-choosable for every integer $m\geq 1$.

#632: [ERT80]

graph theory | chromatic number

A problem of Erdős, Rubin, and Taylor [ERT80]. Note that $G$ is $(a,1)$-choosable corresponds to being $a$-choosable, that is, the list chromatic number satisfies $\chi_L(G)\leq a$.

This is false: Dvořák, Hu, and Sereni [DHS19] construct a graph which is $(4,1)$-choosable but not $(8,2)$-choosable.

See also the entry in the graphs problem collection.

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Additional thanks to: David Penman and Raphael Steiner

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