First we state a few simple finite problems . Chromatic graphs attracted the attention of mathematicians first of all because of the four colour problem, but it was soon realized that interesting and difficult questions can be proved on chromatic graphs which have nothing to do with the four colour problem (or Appel and Haken theorem) . Tutte (and independently Ungar, Zykov and Mycielski) prove...

It is proved that the axiom of determinateness of Mycielski and Steinhaus for games in which players alternate in writing reals implies that co -» (co)" (i-e. for every partition of infinite sets of natural numbers into two classes there is an infinite set such that all its infinite subsets belong to the same class). For every infinite N C co, fi(A/) denotes the family of infinite subsets of N....

Let $(*) denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). Theorem 1. There is a homeomorphism of the unit square onto itself mapping a given set in tyji?) onto a set of Lebesgue measure zero. Theorem 2. There is a set belonging to both Í» and * that cannot be mapped onto a se...

This informal set of notes contains some of the new results which will appear in the author’s 2013 dissertation. We show that every infinite product of locally compact non-compact groups decomposes into the disjoint union of a Haar null set and a meager set, which gives a partial positive answer to a question of Darji. We also show that the compact sets in each such product group are always Haa...

We analyse the entropy of Hermite polynomials and orthogonal polynomials for the Freud weights w(x) = exp(−|x|) on R and show how these entropies are related to information entropy of the one-dimensional harmonic oscillator. The physical interest in such entropies comes from a stronger version of the Heisenberg uncertainty principle, due to Bialynicki-Birula and Mycielski, which is expressed as...

Let P be a perfect subset of the real line, and let the «-element subsets of P be partitioned into finitely many classes, each open (or just Borel) in the natural topology on the collection of such subsets. Then P has a perfect subset whose «-element subsets he in at most (n — 1)! of the classes. Let C be the set of infinite sequences of zeros and ones, topologized as the product of countably m...

The maximality of Abelian subgroups play a role in various parts of group theory. For example, Mycielski [8, 7] has extended a classical result of Lie groups and shown that a maximal Abelian subgroup of a compact connected group is connected and, furthermore, all the maximal Abelian subgroups are conjugate. For finite symmetric groups the question of the size of maximal Abelian subgroups has be...

For a graph G and its spanning tree T the backbone chromatic number, BBC(G,T ), is defined as the minimum k such that there exists a coloring c : V (G) → {1, 2, . . . , k} satisfying |c(u) − c(v)| ≥ 1 if uv ∈ E(G) and |c(u)− c(v)| ≥ 2 if uv ∈ E(T ). Broersma et al. [1] asked whether there exists a constant c such that for every triangle-free graphG with an arbitrary spanning tree T the inequali...

A general theorem is proved which implies that the types of PA (Peano Arithmetic), ZF (Zermelo-Fraenkel Set Theory) and GB (Gödel-Bernays Set Theory) are prime in the lattice of interpretability types. 0. Introduction. One of the important goals of mathematical logic is to investigate the strength of theories. A good approximation to the intuitive concept of strength is the quasiordering "7 is ...

With each of the classical tree-like forcings adjoining a new real, one can associate a σ-ideal on the reals in a natural way. For example, the ideal s0 of Marczewski null sets corresponds to Sacks forcing S, while the ideal r0 of nowhere Ramsey sets corresponds to Mathias forcing R. We show (in ZFC) that none of these ideals is included in any of the others. We also discuss Mycielski’s ideal P...