For every h >= 4 there exists a (2h-1)-uniform hypergraph with no polychromatic 3-coloring whose h-heavy restricted subhypergraphs are all 2-colorable, obtained as the complement of a suitable h-uniform hypergraph on 3h-1 vertices.
P´ alv¨ olgyi, Note on polychromatic coloring of hereditary hypergraph families, Graphs and Combina- torics 40 (2024), article 131
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Note on polychromatic coloring of hereditary hypergraph families II
For every h >= 4 there exists a (2h-1)-uniform hypergraph with no polychromatic 3-coloring whose h-heavy restricted subhypergraphs are all 2-colorable, obtained as the complement of a suitable h-uniform hypergraph on 3h-1 vertices.