Abstract We present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$ .

Abstract We study the number of normal measures on a tall cardinal. Our main results are that: • The least tall cardinal may coincide with the least measurable cardinal and carry as many normal measures as desired. • The least measurable limit of tall cardinals may carry as many normal measures as desired.

Abstract Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at ${\kappa ^{ + + }}$ , assuming that $\kappa = {\kappa ^{ < \kappa }}$ and there is a weakly compact cardinal aboveκ. If in additionκis supercompact then we can forceκto be ${\aleph _\omega }$ in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a ${\kappa ^{ + + }}$ -Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into ${\aleph _\omega }$ .

We prove that it is consistent that $\aleph_\omega$ is strong limit, $2^{\aleph_\omega}$ is large and the universality number for graphs on $\aleph_{\omega+1}$ is small. The proof uses Prikry forcing with interleaved collapsing.

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then 1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W. 2. W weakly covers V if W computes the successor of every V-singular cardinal correctly. Strong covering implies weak covering. In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include 1. X is “0# exists”, Kx is L, Y is “strongly”. 2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”. 3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

Abstract We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen. Square on singular cardinals.

Classical Ramsey theory(at least in its simplest form) is concerned with problems of the following kind: given a setXand a colouring of the set[X]nof unorderedn-tuples fromX, find a subsetY ⊆ Xsuch that all elements of[Y]nget the same colour. Subsets with this property are calledmonochromaticorhomogeneous, and a typical positive result in Ramsey theory has the form that whenXis large enough and the number of colours is small enough we can expect to find reasonably large monochromatic sets. Polychromatic Ramsey theoryis concerned with a “dual” problem, in which we are given a colouring of[X]nand are looking for subsetsY ⊆ Xsuch that any two distinct elements of[Y]ngetdifferentcolours. Subsets with this property are calledpolychromaticorrainbow. Naturally if we are looking for rainbow subsets then our task becomes easier when there are many colours. In particular given an integerkwe say that a colouring isk-boundedwhen each colour is used for at mostkmanyn-tuples. At this point it will be convenient to introduce a compact notation for stating results in polychromatic Ramsey theory. We recall that in classical Ramsey theory we write to mean “every colouring of[κ]ninkcolours has a monochromatic set of order type α”. We will write to mean “everyk-bounded colouring of[κ]nhas a polychromatic set of order type α”. We note that whenκis infinite andkis finite ak-bounded colouring will use exactlyκcolours, so we may as well assume thatκis the set of colours used.

This note proves two theorems. The first is that it is consistent to have for every n, but not have . This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ωn. The crux of the proof is that in the resulting model every stationary subset of ℵω+1 ⋂ cof(ω) reflects to an ordinal of cofinality ω1, that is to say it has stationary intersection with such an ordinal. This result contrasts with compactness properties of square shown in [3]. In that paper it is shown that if one has square at every ωn, then there is a square type sequence on the points of cofinality ωk, k > 1 in ℵω+1. In particular at points of cofinality greater than ω1 there is a strongly non-reflecting stationary set of points of countable cofinality. The second result answers a question of Džamonja, by showing that there can be no squarelike sequence above a supercompact cardinal, where “squarelike” means that one replaces the requirement that the cofinal sets be closed and unbounded by the requirement that they be stationary at all points of uncountable cofinality.

Abstract We prove some results related to the problem of blowing up the power set of the least measurable cardinal. Our forcing results improve those of [1] by using the optimal hypothesis.

Abstract Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals k1..…kn are so that ki; for i = 1..…n is both the ith measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

Abstract We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebraBhas an irredundant subsetAsuch that 2∣A∣= 2∣B∣. This implies in particular thatBhas 2∣B∣subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a “black box” at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere.

Abstract We analyse the Mitchell ordering in a model where κ is -hypermeasurable and > .

In his paper [7] Steel asked whether there can exist a normal measure U on a cardinal κ such that We use Reverse Easton forcing to show that this is consistent from a P2κ hypermeasure; we also show that the result is sharp, using the core model for nonoverlapping coherent extender sequences. The proof uses forcing technology due to Woodin. In this section we collect some facts that are useful in the forcing constructions of the next section. None of them are due to us, and we are unsure to whom they should be attributed for the most part. We give sketchy proofs; the reader who wants to see more details is referred to [2].