Selective ultrafilters and 𝜔→(𝜔)^{𝜔}

Abstract

Mathias (Happy families, Ann. Math. Logic. 12 (1977), 59–111) proved that, assuming the existence of a Mahlo cardinal, it is consistent that CH holds and every set of reals in L ( R ) L(\mathbb {R}) is U \mathcal {U} -Ramsey with respect to every selective ultrafilter U \mathcal {U} . In this paper, we show that the large cardinal assumption cannot be weakened.