In this paper graphs with uncountable chromatic numbers will be studied. As usual, a graph is an ordered pair G = ( V , E), where V is an arbitrary set (the set of vertices), E is a set of unordered pairs from V (the set of edges). A function : V + x (n a cardinal), is a good coloring of G if and only if f ( x ) ¢ f ( y ) whenever x and y are jo ined i.e. joined vertices get different colors. T...

We give a simple, direct, and constructive proof of memoryless determinacy for parity and mean payo& games. First, we prove by induction that the 8nite duration versions of these games, played until some vertex is repeated, are determined and both players have memoryless winning strategies. In contrast to the proof of Ehrenfeucht and Mycielski, Internat. J. Game Theory, 8 (1979) 109–113, our pr...

1 I n t r o d u c t i o n Let G = (V, E) be a finite directed graph in which each vertex has at least one edge going out of it. Let w : E , { W , . . . , 0 , . . . , W} be a function that assigns an integral weight to each edge of G. Ehrenfeucht and Mycielski [EM79] studied the following infinite two-person game played on such a graph. The game starts at a vertex a0 E V. The first player choose...

It is shown that the first-order theory T of a non-surjective injection (of the universe into itself) is connected. By a criterion for connectedness, due to J. Mycielski, one reduces this result to the following theorem: If A, B are any structures and ε is a definable equivalence relation on their disjoint product A ×̇B such that (A ×̇B)/ε is a model of T, then either A has the property that for ...

In this paper we introduce a logic for graph, where the domain is the edges, not the vertices. Such logic behaves differently from the usual graph logic, where the domain is the vertices, because reachability can be expressed in monadic NP in this logic, while it cannot in the standard graph logic. We establish that in this logic to express a path of length 2k or less, quantifier rank k is nece...

The Wehrl entropy is an associated to the Husimi quasi-probability distribution. We discuss how it can be used formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. show that Wehrl-Lieb inequality closer equality than usual Bia{\l}ynicki-Birula Mycielski relation almost everywhere. Furthermore, we mutual information obtain measurable perfect ...

Ando, Matsuzawa, Thom, and Törnquist have resolved a problem by Sorin Popa constructing an example of Polish group unitary operators with the strong operator topology, whose left right uniform structures coincide, but which does not embed into