a b s t r a c tThe structure of ordinals of the form ω ω β for countable β is studied. The main result is: Theorem 1. If β < ω 1 is the sum of one or two indecomposable ordinals, thenAlso an example is given to show that α → (α, 3) 2 need not imply α → (α, n) 2 for all n < ω.
Two new topological partition relations are proved. These are ω 1 → (top α + 1) 2 k and R → (top α + 1) 2 k for all α < ω 1 and all k < ω. Here the prefix "top" means that the homogeneous set α + 1 is closed in the order topology. In particular, the latter relation says that if the pairs of real numbers are partitioned into a finite number of classes, there is a homogeneous (all pairs in the same class), well-ordered subset of arbitrarily large countable order type which is closed in the usual topology of the reals. These relations confirm conjectures of Richard Laver and William Weiss, respectively. They are a strengthening of the classical Baumgartner-Hajnal theorem.
It is shown that the group of almost automorphisms of Rado's 'random graph' cannot be embedded (via a permutation group embedding) into the group of homeomorphisms to itself of the space of rational numbers.
We study a family of infinite games with imperfect information introduced by B. Model for two players that alternately remove and add points to a finite set. We investigate the existence of imperfect information strategies for the remover for different ambient cardinalities. We also study a variant of a game of D. Gale introduced by Scheepers and Weiss.
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