We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras. keywords: generalized graph colorings, compactness, prime ideal theorem MSC: 05C15; 03E25 .

1 Contents Applications of Hamilton's Compactness Theorem for Ricci flow Peter Topping 1 Applications of Hamilton's Compactness Theorem for Ricci flow 3 Overview 3 Background reading 4 Lecture 1. Ricci flow basics – existence and singularities 5 1.1. Initial PDE remarks 5 1.2. Basic Ricci flow theory 6 Lecture 2. Cheeger-Gromov convergence and Hamilton's compactness theorem 9 2.1. Convergence a...

2 Open and Closed Classes in Cantor Space 5 2.1 Open Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Closed Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 The Compactness Theorem . . . . . . . . . . . . . . . . . . . 7 2.4 Notation For Trees . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Effective Compactness Theorem . . . . . . . . . . . . . . . . . 8

Darbo's fixed point theorem and its generalizations play a crucial role in the existence of solutions in integral equations. Meir-Keeler condensing operators is a generalization of Darbo's fixed point theorem and most of other generalizations are a special case of this result. In recent years, some authors applied these generalizations to solve several special integral equations and some of the...

we present some model theoretic results for {l}ukasiewiczpredicate logic by using the methods of continuous model theorydeveloped by chang and keisler.we prove compactness theorem with respect to the class of allstructures taking values in the {l}ukasiewicz $texttt{bl}$-algebra.we also prove some appropriate preservation theorems concerning universal and inductive theories.finally, skolemizatio...

We show that a strong form of the so called Lindstrr om's Theorem 4] fails to generalize to extensions of L ! and L : For weakly compact there is no strongest extension of L ! with the (;)-compactness property and the LL owenheim-Skolem theorem down to. With an additional set-theoretic assumption, there is no strongest extension of L with the (;)-compactness property and the LL owenheim-Skolem ...

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ: For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the LöwenheimS...

Let $X$ be an infinite set, equipped with a topology $tau$. In this paper we studied the relationship between $tau$, and ultrafilters on $X$. We can discovered, among other thing, some relations of the Robinson's compactness theorem, continuity and the separation axioms. It is important also, aspects of communication between mathematical concepts.