Ultrafilters, Transversals, and the Hat Game

Geschke, Lubarsky, and Rahn in ``Choice and the Hat Game’'~\cite{choice-and-the-hat-game} generalize the classic hat game puzzle to infinitely-many players and ask whether every model of set theory without choice in which the optimal solution can be carried out contains either a nonprincipal ultrafilter on $\mathbb{N}$ or else a Vitali set. A negative answer is obtained here by constructing a model in which there is an optimal solution to the hat game puzzle but no nonprincipal ultrafilter on $\mathbb{N}$ and no Vitali set. This is accomplished in a more general setting, establishing that for any Borel bipartite graph $\mathrm{\Gamma }$ not embedding some $K_{n,{\omega }_1}$ and with countable colouring number there is a model of $ZF+DC$ in which $\mathrm{\Gamma }$ has a $2$-colouring but there is no ultrafilter as above or Vitali set. The same conclusion applies to the natural generalization of the hat game to an arbitrary finite number of hat colours.