, By Claim 5.11, adjacency in G J is transitive. This implies the first assertion. The second follows from Claim 5
, Let (A, B) be a Bernstein partition of F with respect
, J into two infinite sets Extend this partition by adding every isolated vertex of G J to A and the rest of J to B. Then extend this partition inductively, level by level, so that every new vertex again has infinite rank in the graph induced by its side. Let this partition of V be denoted by (V A , V B ) By construction of (V A , V B ), the vertices of finite rank in G[V A ] are only those in A. Hence the vertices of G[V A ] that have infinitely many in-neighbours of finite rank are precisely those in A be the undirected graph on J A defined in analogy to G J ; then two vertices from J A are adjacent in G[V A ] J A if and only if they are adjacent in G J . The same remarks apply, with the analogous definitions, By Claim, vol.51113
, By Claim 5.2, u ? ? v ? has a subset X that spans an ?-tournament in G. Assume that u has rank at most the rank of v, and let G be obtained from G by deleting all vertices of smaller rank except those in X
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