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An equivalence between the maximum load problem to a density variant of arithmetic Kakeya sets and the classical multiply-shift hash family of Dietzfelbinger, Hagerup, Katajainen, and Penttonen is made.
This work generalizes and combines tools from the $(k+2)-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023] and yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes.
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