Higher Suslin trees have become a tool in some forcing constructions in set theory (see, for example, [D1] and [D2]). Most of the constructions using ω1 Suslin trees can be extended to κ+ Suslin trees for any regular cardinal κ. Some of these are given in §1.
In many such constructions, sequences of Suslin trees are used. In §II we show, in various ways, that the generalization to sequences, even ω-sequences, of κ+ Suslin trees cannot be done.
In these constructions the Suslin trees are used as forcing poset (the forcing adds a branch in the tree). There is another way to kill a Suslin tree, namely by adding a big antichain. Some results on this forcing are given in §III.
Our notation is standard. If T is a tree and x ∈ T, then ∣x∣ is the height of x in T. We define Tα (or T(α)) = {x ∈ T: ∣x∣ = α} and T∣α = {x ∈ T: ∣x∣ < α}.
If p and q are forcing conditions, p ≤ q means that p has more information than q.
If (Tα: α ∈ I) is a sequence of trees, Π Tα will always mean the set of (xα: α ∈ I) such that xα ∈ Tα and ∣xα∣ = ∣xβ∣ for α, β ∈ I.
For functions b, T,… we denote by b ∣α, T∣ α,… their restriction to α.
If x is a sequence of ordinals and α is an ordinal, x∧α is the sequence obtained by concatenating α at the end of x.