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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)

- E. C. MILNER
- 2004

A tournament .T = <T,-+> is a relational structure on the non-empty set T such that for x, y e T exactly one of the three relations x->y,x=y,y->x holds. Here x-+ y expresses the fact that {x, y} e-+ and we sometimes write this in the alternative form y <-x. Extending the notation to subsets of T we write A-+ B or B <-A if a-+ b holds for all pairs a, b with… (More)

- P Erdös, E C Milner, R Rado
- 2004

For every ordinal number a, an ordered set S is called an rl,,-set if the following condition P,, is satisfied : if A and B are subsets of S, each of cardinal number less than sfi " and if a < b whenever a e A and b e B, then there exists x e S such that a < x for all a e A and x < b for all b e B. rl,,-sets were first introduced and studied by Hausdorff… (More)

We prove that if P is a partial order and P → (ω) 1 ω , then (a) P → (ω + ω + 1, 4) 3 , and (b) P → (ω + m, n) 3 for each m, n < ω. Together these results represent the best progress known to us on the following question of P. Erd˝ os and others. If P → (ω) 1 ω , then does P → (α, n) 3 for each α < ω 1 and each n < ω?