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 ...

The number of variables used by a first-order query is a fundamental measure which has been studied in numerous contexts, and which is known to be highly relevant to the task of query evaluation. In this article, we study this measure in the context of existential positive queries. Building on previous work, we present a combinatorial quantity defined on existential positive queries; we show th...

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G)+1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chr...

We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality א2 such that the second player has a winning strategy in the Ehrenfeucht-Fräıssé-game of length ω1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by χ∗(G). Later, Zhu called it circular chromatic number and denoted it by χc(G). Let χ(G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then χc(G)=χ(G). Denote by M(G) the Mycielski graph of G. Recursively define Mm(G)=...

We consider repetitions in words and solve a longstanding open problem about the relation between the period of a word and the length of its longest unbordered factor (where factor means uninterrupted subword). A word u is called bordered if there exists a proper prefix that is also a suffix of u, otherwise it is called unbordered. In 1979 Ehrenfeucht and Silberger raised the following problem:...

The authors show, by means of a finitary version ¤ λ,D of the combinatorial principle ¤ ∗ λ of [7], the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal λ, if Mi and Ni are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fräıssé g...

We study how equivalent nonisomorphic models of unsuperstable theories can be. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues [HS].

We study how equivalent nonisomorphic models an unsuperstable theory can have. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues the work started in [HT].