Karel Prikry

Unfriendly partitions of a graph

Minnesota, Minneapoli.~, Minnesota Communicated by the Managing Editors Received March 8. 1988 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 finite graph. We show that the conjecture is also true for graphs which satisfy one of the following two conditions: (i) there are only finitely many vertices having infinite degrees; (ii) there are a finite number of infinite cardinals “to < ntI < cm, such that m, is regular for 1 < i 6 X-, there are fewer than m,, vertices having finite degrees, and every vertex having infinite degree has degree m, for some i < k.

On Partitions into Stationary Sets

We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, V = L . We shall show, among other things, that if V = L then for each cardinal λ there is a set A ⊆ λ such that neither A nor λ – A contain a closed set of type ω 1 . This is an extension of a result of Silver who proved it for λ = ω 2 , providing a partial answer to Problem 68 of Friedman [2]. The main results of this paper were obtained independently by both authors. If λ is an ordinal, E is said to be Mahlo (or stationary) in λ, if λ – E does not contain a closed cofinal subset of λ. Consider the statements: (J 1 ) There is a class E of limit ordinals and a sequence C λ defined on singular limit ordinals λ such that (i) E ⋂ μ is Mahlo in μ for all regular > ω; (ii) C λ is closed and unbounded in λ; (iii) if γ C λ , then γ is singular, γ ∉ E and C γ = γ ⋂ C λ . For each infinite cardinal κ: (J 2,κ ) There is a set E ⊂ κ + and a sequence C λ (Lim(λ), λ + ) such that (i) E is Mahlo in κ + ; (ii) C λ is closed and unbounded in λ; (iii) if cf(λ) C λ (iv) if γ C λ then γ ∉ E and C γ = ϣ ⋂ C λ .

The locally finite topology on 2

Let X be a metrizable space. A Vietoris-type topology, called the locally finite topology, is defined on the hyperspace 2x of all closed, nonempty subsets of X. We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on X. We also investigate when the locally finite topology coincides with the more usual topologies on 2x and when the locally finite topology is metrizable.

Caratheodory-type selections and random fixed point theorems☆

Abstract We provide some new Caratheodory-type selection theorems, i.e., selections for correspondences of two variables which are continuous with respect to one variable and measurable with respect to the other. These results generalize simultaneously Michaels [21]continuous selection theorem for lower-semicontinuous correspondences as well as a Caratheodory-type selection theorem of Fryszkowski [10]. Random fixed point theorems (which generalize ordinary fixed point theorems, e.g., Browders [6]) follow as easy corollaries of our results.

THE MEASURABILITY OF UNCOUNTABLE UNIONS

(1.1) Null sets are unimportant and can be ignored. (1.2) Nonmeasurable sets play the role of black sheep in an old established family. One grudgingly acknowledges their existence on special occasions, and otherwise one tries not to think about them. (1.3) Consequently, uncountable Boolean operations on measurable sets are taboo, since they might bring the scandalous black sheep into full view. (1.4) The Lusin criterion of measurability provides transcendent insight into the true nature of a measurable function, but is otherwise useless. (1.5) Its manifold applications notwithstanding, measure theory is stone cold dead as a viable area of current research.

The hausdorff metric and measurable selections

Abstract We construct measurable selections for closed set-valued maps into arbitrary complete metric spaces. We do not need to make any separability assumptions. We view the set-valued maps as point-valued maps into the hyperspace and our measurability assumptions arethe usual kinds of measurability of point-valued maps in this setting. We also discuss relationship of these measurability conditions to the ones usually considered in the theory of measurable selections.

On a theorem of silver

A recent theorem of Silver, in its simplest form, states, that if @w < cf(k) < k and 2^@l=@l^+ for all @l < k, then 2^k=k^+. Silvers proof employs Boolean-valued as well as nonstandard models of set theory. In the present note we give an elementary proof of Silvers theorem in its general form.

Ultrafilters and almost disjoint sets

Abstract Let μκ denote the space of uniform ultrafilters on κ, and let λ be a cardinal. U ϵμκ is said to be a λ-point of μκ if U is a boundary point of λ pairwise disjoint open subsets of μκ. We prove that if κ is a successor cardinal, 2 κ = κ + , and Kurepas hypothesis for κ holds, then each U ϵ μκ is a 2 κ -point of μκ.

A partition relation for triples using a model of Todorccevic

TodorÂ?eviÂ? has shown that there is a ccc extension M in which MAÂ?1 + 2Â? = Â?2 holds and also in which the partition relation Â?i Â? (Â?1,Â?)2 holds for every denumerable ordinal Â?. We show that the partition relation for triples Â?1 Â? (Â?2 + 1, 4)3 holds in the model M, and hence by absoluteness this is a theorem in ZFC.

Measurability of the value of a parametrized game

Results from descriptive set theory are used to study measurability properties of the (upper) value of a measurably parametrized family of two-person, zero-sum games with measurable payoffs.