The Dichotomy Theorems

1. The G0 dichotomy A digraph (or directed graph) on a set X is a subset G ⊆ X \ ∆. Given a digraph G on a set X and a subset A ⊆ X, we say that A is G-discrete if for all x, y ∈ A we have (x, y) / ∈ G. Now let sn ∈ 2 be chosen for every n ∈ N such that ∀s ∈ 2<N ∃n s v sn. Then we can define a digraph G0 on 2N by G0 = {(sn0x, sn1x) ∈ 2N × 2N ∣∣ n ∈ N & x ∈ 2N}. Lemma 1. If B ⊆ 2N has the Baire property and is non-meagre, then B is not G0-discrete. Proof. By assumption on B, we can find some s ∈ 2<N such that B is comeagre in Ns. Also, by choice of (sn), we can find some n such that s v sn, whereby B is comeagre in Nsn . By the characterisation of comeagre subsets of 2 N, we see that for some x ∈ 2N, we have sn0x, sn1x ∈ B, showing that B is not G0-discrete.