By James E. Baumgartner, András Hajnal (auth.), N. W. Sauer, R. E. Woodrow, B. Sands (eds.)

This quantity includes the money owed of papers brought on the Nato complicated examine Institute on Finite and endless Combinatorics in units and good judgment held on the Banff Centre, Alberta, Canada from April 21 to may possibly four, 1991. because the identify indicates the assembly introduced jointly staff attracted to the interaction among finite and limitless combinatorics, set conception, graph conception and common sense. It was that endless set thought, finite combinatorics and common sense will be considered as rather separate and self sustaining matters. yet an increasing number of these disciplines develop jointly and turn into interdependent of one another with ever extra difficulties and effects showing which problem all of these disciplines. I delight in the monetary aid which used to be supplied by way of the N. A. T. O. complex learn Institute programme, the normal Sciences and Engineering examine Council of Canada and the dep. of arithmetic and information of the college of Calgary. 11l'te assembly on Finite and limitless Combinatorics in units and good judgment different conferences on discrete arithmetic held in Banff, the Symposium on Ordered units in 1981 and the Symposium on Graphs and Order in 1984. The turning out to be inter-relation among the various parts in discrete arithmetic is probably top illustrated by way of the truth that a few of the contributors who have been current on the past conferences additionally attended this assembly on Finite and endless Combinatorics in units and Logic.

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Em satisfying m L c(ej) = O. ;=1 This paper is of expository nature. In Section 2, we shall review some generalizations by Z. Fiiredi & D. Kleitman, A. Schrijver & P. Seymour, and A. Bialostocki & P. Dierker. In Section 3, we shall introduce the notion of a proper Ramsey type theorem and make a few observations. Section 4 contains a collection of fifteen conjectures concerning zero sum problems which involve trees and forests. 2 The zero-sum tree theorem and its generalizations We start with some definitions.

B) Describe the superatomic CO algebras. 23(a). Let B be a Boolean algebra. We recall that a subset S of B is dense in B if for every OB 01= b E B, there is a E S such that 0 < a::; b. Let d(B) = min({ISI : B has a dense set S ~ B}). Let (A*) denote the following axiom of set theory: for every infinite ordinal a, 2101 > No. 24 (Shelah) Assume A*. Let B be a CO algebra. For nEw, let Un(B) be the 0 set of a E B such that d(Br a) < Nn . Then for some nEw, Un(B) is dense in X. Note that for every n < w, Un(B) is an ideal of B, and B is atomic if and only if Uo(B) is dense in B.

The definition of CO space is related to the following notion. e. a Hausdorff space). We say that X is a Toronto space if every subspace of X of the same cardinality, is homeomorphic to X. For example, there is only one countable Toronto space (up to homeomorphy), namely the discrete topology on a countable set. We don't know if there an uncountable non-discrete Toronto space. Every ordinal algebra is a CO algebra, and because every homomorphic image of such an algebra is an ordinal algebra, such an algebra is a HCO algebra.