Antichains of perfect and splitting trees

Tree Forcing and Definable Maximal Independent Sets in Hypergraphs

“…We will also prove the case of finite products but our main focus will be on the countable support iteration. Splitting forcing (Definition 4.1) is a less-known forcing notion that was originally introduced by Shelah in [27] and has been studied in more detail recently [17, 20, 29, 30]. Although it is very natural and gives a minimal way to add a splitting real (see more below), it has not been exploited a lot and to the best of our knowledge, there is no major set theoretic text treating it in more detail.…”

Tree forcing and definable maximal independent sets in hypergraphs

“…We will also prove the case of finite products but our main focus will be on the countable support iteration. Splitting forcing SP (Definition 4.1) is a less-known forcing notion that was originally introduced by Shelah in [22] and has been studied in more detail recently ( [24], [25], [13] and [16]). Although it is very natural and gives a minimal way to add a splitting real (see more below), it has not been exploited a lot and to our knowledge, there is no major set theoretic text treating it in more detail.…”

A Sacks amoeba preserving distributivity of $\mathcal {P}(\omega )/\mathrm {fin}$

“…This can be seen as follows. Let κ(S) denote (following the notation in [HS20]) the least cardinal to which Sacks forcing collapses the continuum (in [Rep08], this cardinal characteristic is denoted by sh(S)). By [HS20, Theorem 2.7], add(s 0 ) ≤ κ(S) holds true in ZFC, and by [Rep08,Lemma 3.5(3)], κ(S) ≤ h ω holds true in ZFC.…”

“…However, the question which of them are upper bounds of add(s 0 ) is interesting and linked to hard open problems. In [HS20], the inequality add(s 0 ) ≤ b was proved. This had been known previously in the case where c is regular by work of [JMS92] and [Sim93].…”

“…An easy explicit proof for our amoeba forcing A is given in Lemma 3.22. A short direct proof of add(s 0 ) ≤ b-see Theorem 2.14(i)-is only available under the assumption a(S) = c (i.e., the assumption that every uncountable maximal antichain in Sacks forcing is of size continuum), so in particular under d = c, due to d ≤ a(S) proved in [HS20].…”