Reviewed Works:Edwin W. Miller, On a Property of Families of Sets.Ben Dushnik, E. W. Miller, Partially Ordered Sets.P. Erdos, Some Set-theoretical Properties of Graphs.G. Fodor, Proof of a Conjecture of P. Erdos.P. Erdos, R. Rado, A Partition Calculus in Set Theory.P. Erdos, R. Rado, Intersection Theorems for Systems of Sets.A. Hajnal, Some Results and Problems on Set Theory.P. Erdos, A. Hajnal, On a Property of Families of Sets.A. Hajnal, Proof of a Conjecture of S. Ruziewicz.P. Erdos, A. Hajnal, R. Rado, (...) Partition Relations for Cardinal Numbers.P. Erdos, A. Hajnal, On a Problem of B. Jonsson.P. Erdos, A. Hajnal, On Chromatic Number of Graphs and Set-Systems. (shrink)

It is shown that if ZF is consistent, then so is ZFC + GCH + "There is a graph with cardinality ℵ 2 and chromatic number ℵ 2 such that every subgraph of cardinality ≤ ℵ 1 has chromatic number ≤ ℵ 0 ". This partially answers a question of Erdos and Hajnal.

Several canonical partition theorems are obtained, including a simultaneous generalization of Neumer's lemma and the Erdos-Rado theorem. The canonical partition relation for infinite cardinals is completely determined, answering a question of Erdos and Rado. Counterexamples are given showing that in several ways these results cannot be improved.

Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ + -saturated, i.e., are there κ + stationary subsets of κ with pairwise intersections nonstationary? Our first observation is: Theorem. NS is κ + -saturated iff for every normal ideal J on κ there is a stationary set $A \subseteq \kappa$ such that $J = NS \mid A = \{X \subseteq (...) \kappa:X \cap A \in NS\}$ . Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define "greatly Mahlo" cardinals and obtain the following: Theorem. If κ is greatly Mahlo then NS is not κ + -saturated. Theorem. If κ is ordinal Π 1 1 -indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a κ-saturated ideal, then κ is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π 1 1 -indescribable cardinal. These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ + -saturated. (shrink)

If dominating functions in ω ω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered unbounded family of increasing functions is a scale, and the splitting number s (and hence the distributivity number h) remains at ω 1.

We study various aspects of the size, including the cardinality, of closed unbounded subsets of [λ]<κ, especially when λ = κ+n for n ε ω. The problem is resolved into the study of the size of certain stationary sets. Relative to the existence of an ω1-Erdös cardinal it is shown consistent that ωω3 < ωω13 and every closed unbounded subsetof [ω3]<ω2 has cardinality ωω13. A weakening of the ω1-Erdös property, ω1-remarkability, is defined and shown to be retained under a large (...) class of Easton-like forcings applied to ω1-Erdös cardinals. A class of reverse-Easton forcings preserving α-Erdösness is also described, with special attention to the establishment of □-principles. Article Outline. (shrink)

We study the I-ultrafilters on ω, where I is a collection of subsets of a set X, usually R or ω 1 . The I-ultrafilters usually contain the P-points, often as a small proper subset. We study relations between I-ultrafilters for various I, and closure of I-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I-ultrafilters always exist.

It is shown that for any cardinal $\kappa, \dbinom{(2^{ , and if κ is weakly compact $\dbinom{\kappa^+}{\kappa} \rightarrow \dbinom{\kappa}{\kappa}_{.