Dense uniform hypergraphs have high list chromatic number

The first author showed that the list chromatic number of every graph with average degree d is at least ( 0.5 − o ( 1 ) ) log 2 d . We prove that for r ≥ 3 , every r -uniform hypergraph in which at least half of the ( r − 1 ) -vertex subsets are contained in at least d edges has list chromatic number at least ln d 100 r 3 . When r is fixed, this is sharp up to a constant factor.