Related papers: The adjacent Hindman's theorem for uncountable gro…

We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every…

Recent results of Hindman, Leader and Strauss and of Fern\'andez-Bret\'on and Rinot showed that natural versions of Hindman's Theorem fail {\em for all} uncontable cardinals. On the other hand, Komj\'ath proved a result in the positive…

A particular case of the Hindman--Galvin--Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums…

Hirst investigated a slight variant of Hindman's Finite Sums Theorem -- called Hilbert's Theorem -- and proved it equivalent over $\RCA_0$ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural…

Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring…

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…

This paper continues a line of investigation of the Halpern--L\"{a}uchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--L\"{a}uchli Theorem for one tree of height $\kappa$ holds whenever $\kappa$ is strongly…

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…

We prove that the existence of a non-special tree of size $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no $K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the special tree number…

Hindman's Theorem asserts that, for each finite coloring of the natural numbers, there are distinct natural numbers $a_1,a_2,\dots$ such that all of the sums $a_{i_1}+a_{i_2}+\dots+a_{i_m}$ ($m\ge 1$, $i_1<i_2<\dots<i_m$) have the same…

We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta<\kappa$, the existence of a strongly unbounded coloring…

We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $\kappa$,…

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly…

We consider the restriction of Ramsey's theorem that arises from considering only translation-invariant colourings of pairs, and show that this has the same strength (both from the viewpoint of Reverse Mathematics and from the viewpoint of…

We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is…

Suppose lambda is a singular cardinal of uncountable cofinality kappa. For a model M of cardinality lambda, let No(M) denote the number of isomorphism types of models N of cardinality lambda which are L_{infty lambda}-equivalent to M. In…

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not…

We prove that for infinite cardinals $\kappa<\lambda$ the alternating group $Alt(\lambda)$ (of even permutations) of $\lambda$ is not embeddable into the symmetric group $Sym(\kappa)$ (of all permutations) of $\kappa$. To prove this fact we…

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…

It is proved that in Godel's constructible universe, for every infinite successor cardinal k, there exist graphs G and H of size and chromatic number k, for which the tensor product graph (G x H) is countably chromatic.