Related papers: Coloring ordinals by reals

A duality of $\kappa$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $\kappa$-normability of a topological vector space…

Working in subsystems of second order arithmetic, we formulate several representations for hypergraphs. We then prove the equivalence of various vertex coloring theorems to ${\sf WKL}_0$, ${\sf ACA}_0$ and $\Pi ^1_ 1$-${\sf CA}_0$.

We consider an inverse spectral problem for a class of singular AKNS operators $H\_a, a\in\N$ with an explicit singularity. We construct for each $a\in\N$, a standard map $\lambda^a\times\kappa^a$ with spectral data $\lambda^a$ and some…

Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$),…

We prove that if the set of unordered pairs of real numbers is colored by finitely many colors, there is a set of reals homeomorphic to the rationals whose pairs have at most two colors. Our proof uses large cardinals and it verifies a…

A set-system $X$ is a $(\lambda, \kappa,\mu)$-system iff $|X|=\lambda$, $|x|=\kappa$ for each $x\in X$, and $X$ is $\mu$-almost disjoint. We write $[\lambda, \kappa, \mu] -> \rho$ iff every $(\lambda, \kappa,\mu)$-system has a "conflict…

We investigate the extent to which ultrapowers by normal measures on $\kappa$ can be correct about powersets $\mathcal{P}(\lambda)$ for $\lambda>\kappa$. We consider two versions of this questions, the capturing property…

Formalised libraries of combinatorial mathematics have rapidly expanded over the last five years, but few use one of the most important tools: probability. How can often intuitive probabilistic arguments on the existence of combinatorial…

In this paper we use infinitary Turing machines with tapes of length $\kappa$ and which run for time $\kappa$ as presented, e.g., by Koepke \& Seyfferth, to generalise the notion of type two computability to $2^{\kappa}$, where $\kappa$ is…

We study the determinacy of the game G_kappa (A) introduced in [FKSh:549] for uncountable regular kappa and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that G_kappa (A) is…

This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{<kappa}, pcf-inaccessibility, entangled orders (and narrow Boolean Algebras),…

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 extend our previous study of Hopf-algebraic $\kappa$-deformations of all inhomogeneous orthogonal Lie algebras ${\rm iso}(g)$ as written in a tensorial and unified form. Such deformations are determined by a vector $\tau$ which for…

Generalizing Keisler's notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on $\omega_1$ must be somewhere $\omega_1$-dense. We prove a dichotomy…

Ordinal Classification (OC) is a widely encountered challenge in Natural Language Processing (NLP), with applications in various domains such as sentiment analysis, rating prediction, and more. Previous approaches to tackle OC have…

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach,…

We study Ramsey's theorem for pairs and two colours in the context of the theory of $\alpha$-large sets introduced by Ketonen and Solovay. We prove that any $2$-colouring of pairs from an $\omega^{300n}$-large set admits an $\omega^n$-large…

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…

The number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis…

Inspired by Owings's problem, we investigate whether, for a given an Abelian group $G$ and cardinal numbers $\kappa,\theta$, every colouring $c:G\longrightarrow\theta$ yields a subset $X\subseteq G$ with $|X|=\kappa$ such that $X+X$ is…