Given any finite alphabet A and positive integers m1, ..., mk, congruences on A*, denoted by ~(m1, ..., mk) and related to a version of the Ehrenfeucht-Fraisse game, are defined. Level k of the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoids A*/~(m1, ... mk). A natural subhierarchy of level 2 and equation systems satisfied in the corresponding varieties of ...

For [Formula: see text], the coarse similarity class of denoted by is set all text] such that symmetric difference and has asymptotic density text]. There a natural metric on space classes defined letting be upper We study under this metric, show in particular between any two distinct points there are continuum many geodesic paths. also subspaces form where closed Turing equivalence, tight conn...

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...

S. Banach pointed out that the graph of generic (in sense Baire category) element $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if measure theoretic dual holds, i.e., all but Haar null many Christensen) elements have We answer this question in affirmative. call $f \in \text{Homeo}([0,1]^d)$ singular it takes a suitable set full to nullset, and strongly is almost everywhere differenti...

Let G be a connected graph with odd girth 2k + 1. Then G is a (2k + 1)-angulated graph if every two vertices of G are joined by a path such that each edge of the path is in some (2k + 1)-cycle. We prove that if G is (2k+1)-angulated, and H is connected with odd girth at least 2k+3, then any retract R of the box (or Cartesian) product G2H is isomorphic to S2T where S is a retract of G and T is a...

The Mycielski ideal Mk is defined to consist of all sets A ⊆ k such that {f ↾ X : f ∈ A} 6= k for all X ∈ [ω]0 . It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Ros lanowski ideals will be shown to be consistently different.