Related papers: Dear Lambda

We present a forcing for blowing up 2^lambda and making ``many positive polarized partition relations'' (in a sense made precise in (c) of our main theorem) hold in the interval [lambda, 2^lambda]. This generalizes results of [276], Section…

We generalize to the relations $(\lambda, \mu) \stackrel{\kappa}{\Rightarrow} (\lambda', \mu')$ and $\alm (\lambda, \mu) \stackrel{\kappa}{\Rightarrow} \alm (\lambda', \mu')$ some results obtained in Parts II and IV. We also present a…

We prove the consistency of $\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{p}=\mathfrak{g}=\mathfrak{s}<\mathrm{add}(\mathcal{M})=\mathrm{cof}(\mathcal{M})<\mathfrak{a}=\mathrm{non}(\mathcal{N})=\mathfrak{c}$ with ZFC where…

Recently it has been proved that, assuming that there is an almost disjoint family of cardinality (2^{\mathfrak c}) in (\mathfrak c) (which is assured, for instance, by either Martin's Axiom, or CH, or even $2^{<\mathfrak c=\mathfrak c$})…

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V models ZFC + GCH is a given model (which in interesting cases contains instances of…

Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either…

Several authors have argued that self-consistent $f(R)$ gravity models distinct from $\Lambda $CDM are almost ruled out. Confronting such claims, we present a particular two-parameter $f(R)$ model that: (a) is cosmologically viable and…

We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize…

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many…

We present two different types of models where, for certain singular cardinals lambda of uncountable cofinality, lambda -> (lambda, omega+1)^2, although lambda is not a strong limit cardinal. We announce, here, and will present in a…

We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…

We present some promising ideas to treat the problem of making completely rigorous the development of our expression for $\lambda_d(p)$ of the monomer-dimer problem on a $d$-dimensional hypercubic lattice \begin{equation}\label{abstract1}…

We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $ \lambda $ is a singular cardinal, $ \lambda ' < \lambda $, and the ultrafilter $D$ is $ \kappa $-decomposable for all…

Let kappa be a regular uncountable cardinal and lambda > kappa a singular strong limit cardinal. We give a new characterization of the nonstationary subsets of P_kappa (lambda) and use this to prove that the nonstationary ideal on P_kappa…

We point out a gap in Shelah's proof of the following result: $\mathbf{Claim}$ Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal $\lambda$ such that whenever $M, N \in K$ have…

In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square(\lambda^+)$. Furthermore, we establish…

We succeed to say something on the identities of (mu^+, mu) when mu>theta>cf(mu), mu strong limit theta--compact. This hopefully will help to prove the consistency of ``some pair (mu^+,mu) is not compact'', however, this has not been…

Let lambda be an infinite cardinal number and let C = {H_i| i in I} be a family of nontrivial groups. Assume that |I|<=lambda, |H_i|<= lambda, for i in I, and at least one member of C achieves the cardinality lambda. We show that there…

We prove the consistency of the failure of the weak diamond $\Phi_\lambda$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $\Psi_\lambda$ is equivalent to the statement $2^{<\lambda}<2^\lambda$ and…

We prove that for any regular kappa and mu > kappa below the first fix point (lambda = aleph_lambda) above kappa, there is a graph with chromatic number > kappa, and mu^kappa nodes but every subgraph of cardinality < mu has chromatic number…