Cardinal characteristics of the continuum and partitions

Abstract

We prove that for regular cardinals κ, combinations of the stick principle at κ and certain cardinal characteristics at κ being κ + causes the partition relations such as ω 1 −→ (ω 1 , ω + 2) 2 and (κ + ) 2 −→ (κ + κ, 4) 2 to fail. Polarised partition relations are also considered, and the results are used to answer several problems posed

Tiltan and Superclub

“…In order to give a negative answer we will show that the positive relation . On the other hand, tiltan implies stick and by Proposition 3.3 of [3] we have κ ω 1 → κ ω 2 and a fortiori κ κ → κ ω 2 . From Theorem 7 we infer that every κ-reasonable function f : κ → P (κ) has an infinite free set, so we are done.…”