Filter‐Menger set of reals in Cohen extensions

Abstract We prove that for every ultrafilter on there exists a filter on which is ‐Menger and . We show that in the Cohen model there exists such which are tall by using a construction of Nyikos's 10. These answer a question of Das 2, Problem 7. We prove that there is a Menger filter of character that is not Hurewicz in the ‐Cohen model where is uncountable regular. This shows that the positive answer to a question of Hernández‐Gutiérrez and Szeptycki 3, Question 2.8 is consistent with . We also study the filter generated by the set of mutually Cohen reals in the ‐Cohen model. We prove that and and every ‐dominating family in the ground model is ‐unbounded in extension. Two questions are posed.