A structure $={(A;E_i)}_{i\in n}$ where each $E_i$ is an equivalence relation on A is called an n-grid if any two equivalence classes coming from distinct $E_i$’s intersect in a finite set. A function χ: A → n is an acceptable coloring if for all i ∈ n, the ${\chi }^{-1}\left(i\right)$ intersects each $E_i$-equivalence class in a finite set. If B is a set, then the n-cube Bⁿ may be seen as an n-grid, where the equivalence classes of $E_i$ are the lines parallel to the ith coordinate axis. We use elementary submodels of the universe to characterize those n-grids...

We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^s\left(\kappa \right)$ and $F^s\left(\kappa \right)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂))...

Let $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ be the set of subsets of $I$ of cardinality $m$. Let $f$ be a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$ and $g$ a coloring of $\left(\genfrac{}{}{0pt}{}{I}{m}\right)$. We write $f\to g$ if every $f$-homogeneous $H\subseteq I$ is also $g$-homogeneous. The least $m$ such that $f\to g$ for some $f:\left(\genfrac{}{}{0pt}{}{I}{m}\right)\to k$ is called the $k$-width of $g$ and denoted by $w_k\left(g\right)$. In the first part of the paper we prove the existence of colorings with high $k$-width. In particular, we show that for each $k>0$ and $m>0$ there is a coloring $g$ with $w_k\left(g\right)=m$. In the second part of the paper we give applications of wide colorings in the theory of generalized quantifiers....

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$ describing properties of Dense(ℚ). These invariants satisfy ${}_{\mathbb{Q}}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$.WecomparethemwiththeiranaloguesinthewellstudiedBooleanalgebra\left(\omega \right)/fin.Weshowthat$ℚ = p$,$ℚ = t$and$ℚ = i$,whereas$ℚ > h$and$ℚ > r$arebothshowntoberelativelyconsistentwithZFC.Wealsoinvestigatecombinatoricsoftheidealnwdofnowheredensesubsetsof,\mathbb{Q}.Inparticular,weshowthat$non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

This note is devoted to combinatorial properties of ideals on the set of natural numbers. By a result of Mathias, two such properties, selectivity and density, in the case of definable ideals, exclude each other. The purpose of this note is to measure the ``distance'' between them with the help of ultrafilter topologies of Louveau.

Some of the covering properties of spaces as defined in Parts I and II are here characterized by games. These results, applied to function spaces $C_p\left(X\right)$ of countable tightness, give new characterizations of countable fan tightness and countable strong fan tightness. In particular, each of these properties is characterized by a Ramseyan theorem.

We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31/2}$-space. In [9] we showed that $C_p\left(X\right)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_p\left(X\right)$ has countable fan tightness and the Reznichenko property. 2....

Using techniques for modeling indices by means of functional equations and resources from fuzzy set theory, the classical Balthazard index used in order to combine several degrees of impairment is characterized in two natural ways and its use is criticized. In addition some hints are given on how to study better solutions than Balthazard's one for the problem of combining impairment degrees.

This paper describes UIO, a multi–domain question–answering system for the Czech language that looks for answers on the web. UIO exploits two fields, namely natural language interface to databases and question answering. In its current version, UIO can be used for asking questions about train and coach timetables, cinema and theatre performances, about currency exchange rates, name–days and on the Diderot Encyclopaedia. Much effort have been made into making addition of a new domain very easy. No...

The concept of a commutative directoid was introduced by J. Ježek and R. Quackenbush in 1990. We complete this algebra with involutions in its sections and show that it can be converted into a certain implication algebra. Asking several additional conditions, we show whether this directoid is sectionally complemented or whether the section is an NMV-algebra.

We study commutative directoids with a greatest element, which can be equipped with antitone bijections in every principal filter. These can be axiomatized as algebras with two binary operations satisfying four identities. A minimal subvariety of this variety is described.