On Club-like Principles on Regular Cardinals above Ω

We prove that for regular λ above a strong limit singular µ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such λ and µ there are coboundedly many regular κ < µ such that ♣ − (S λ κ) holds whenever λ = λ <κ. 1 0 Introduction. The main result of this note is that for any regular cardinal λ above ω there are unboundedly many regular cardinals κ < ω such that provided λ = λ <κ , the combinatorial principle ♣ − (S λ κ) holds. That principle is defined in the following Definition 0.1, and the notation S λ κ is recalled in 0.2. 1 The author thanks EPSRC for their support through the grant number GR/M71121. She is also grateful to Prof. Balcar and Prof. Simon of the Charles University in Prague for their invitation to speak at the 29th Winter School of Abstract Analysis, which proved to be a most productive and pleasant experience.