Three-colorability of I0-avoiding 2→1 hypergraphs

Determine whether every 2→1 hypergraph that does not contain the two-edge hypergraph I_0 as a subhypergraph admits a proper 3-coloring.

Background

The paper proves that avoiding I_0 guarantees a proper 4-coloring for 2→1 hypergraphs. It also constructs a 2→1 hypergraph avoiding I_0 whose chromatic number is three, showing that two colors do not always suffice.

The authors therefore leave unresolved whether the upper bound can be improved from four colors to three colors for the entire class of I_0-avoiding 2→1 hypergraphs.