In 1992, A. Ehrenfeucht and J. Mycielski defined a seemingly pseuorandom binary sequence which has since been termed the EM-sequence. The balance conjecture for the EM-sequence, still open, is the conjecture that the sequence of EM-sequence initial segment averages converges to 1=2. In this paper, we do not prove the balance conjecture but we do make some progress concerning it, namely, we prov...

What largeness and structural assumptions on A ⊆ [R]ω can guarantee the existence of a non-empty perfect set P ⊆ R such that [P ]ω ⊆ A? Such a set P is called A-homogeneous. We show that even if A is open, in general it is independent of ZFC whether for a cardinal κ, the existence of an A-homogeneous set H ∈ [R]κ implies the existence of a non-empty perfect A-homogeneous set. On the other hand,...

We study the complexity of nding the values and optimal strategies of mean payoo games on graphs, a family of perfect information games introduced by Ehrenfeucht and Mycielski and considered by Gurvich, Karzanov and Khachiyan. We describe a pseudo-polynomial time algorithm for the solution of such games, the decision problem for which is in NP \ co-NP. Finally, we describe a polynomial reductio...

∗BASED ON A TALK PRESENTED AT “NEW INSIGHTS IN QUANTUM MECHANICS – FUNDAMENTALS, EXPERIMENTAL RESULTS, THEORETICAL DIRECTIONS”, GOSLAR, AUGUST 31 – SEPTEMBER 3, 1998. Let us mention just some of them: T.W.B. Kibble , A. Ashtekar and T.A. Schilling , P. Bona , R. Haag and U. Bannier , Mielnik , S. Weinberg , G. Auberson and P.C. Sabatier , M.D. Kostin , H.D. Doebner and Goldin , I. Bialynicki-Bi...

(compacta) and continuous mappings was founded by Shchepin [Sh]. He described some elementary properties of such functors and defined the notion of the normal functor which has become very fruitful. The classes of all normal and weakly normal functors include many classical constructions: the hyperspace exp, the space of probability measures P, the superextension λ , the space of hyperspaces of...

The paper shows how the principle that the whole must be greater than the part is not necessarily inconsistent with being bijected with a proper subset, provided that equicardinality is reinterpreted as related with definability and not with sameness of size. An explanation for such reinterpretation is offered on the basis of availability, which leads to the problem of graduality, as raised by ...

We prove that the coindex of the box complex B(H) of a graph H can be measured by the generalised Mycielski graphs which admit a homomorphism to it. As a consequence, we exhibit for every graph H a system of linear equations solvable in polynomial time, with the following properties: If the system has no solutions, then coind(B(H)) + 2 ≤ 3; if the system has solutions, then χ(H) ≥ 4.

The well-known Zarankiewicz problem [Za] is to determine the least positive integer Z(m,n, r, s) such that each m × n 0-1 matrix containing Z(m,n, r, s) ones has an r × s submatrix consisting entirely of ones. In graph-theoretic language, this is equivalent to finding the least positive integer Z(m, n, r, s) such that each bipartite graph on m black vertices and n white vertices with Z(m,n, r, ...

We consider a number of related results taken from two papers – one by W. Lin [1], and the other D. C. Fisher[2]. These articles treat various forms of graph colorings and the famous Mycielski construction. Recall that the Mycielskian μ(G) of a simple graph G is a graph whose chromatic number satisfies χ(μ(G)) = χ(G)+1, but whose largest clique is no larger than the largest clique in G. Extendi...

In this paper, we survey some of the results and open questions regarding colorings of infinite graphs. We first introduce terminology to describe finite and infinite colorings, then offer a proof of the De Bruijn-Erdős Theorem, one of the primary tool for studying the chromatic number of infinite graphs. We show how the De Bruijn-Erdős Theorem allows us to extend results of finite graphs of ar...