E. C. Milner

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It has been conjectured by Cowan and Emerson [3] that every graph has an unfriendly partition; i.e., there is a partition of the vertex set V= V, v V, such that every vertex of V, is joined to at least as many vertices in V, _, as to vertices in V,. It is easily seen that every rinite graph has such a partition, and hence by compact-ness so does any locally(More)
We answer a variant of a question of Rödl and Voigt by showing that, for a given infinite cardinal λ, there is a graph G of cardinality κ = (2 λ) + such that for any colouring of the edges of G with λ colours, there is an induced copy of the κ-tree in G in the set theoretic sense with all edges having the same colour.