Martin Ziegler

This paper is a study of topological lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lovasz in 1978, in his famous proof of the Kneser conjecture via Algebraic Topology. This conjecture stated that the Kneser graph KGm,n ,t he graph with allk-element subsets of {1,2,...,n} as vertices and all pairs of disjoint sets as edges, has chromatic number n! 2k+2. Several other proofs have since been published (by Barany, Schrijver, Dol'nikov, Sarkaria, Kr´iz,

As it is well known, the groups of definable automorphisms of two elementary equivalent relational structures satisfy the same V,-statements. We show that this does not hold in general for V 2 -statements 9 thus correcting an error in the literature.

By the sometimes so-called Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS2000), Brattka (MLQ2005), and Ziegler (ToCS2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the jump of a representation: both a TTE–counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical

Abstract For regular sets in Euclidean space, previous work has identified twelve ‘basic’ computability notions to (pairs of) which many previous notions considered in literature were shown to be equivalent. With respect to those basic notions we now investigate on the computability of natural operations on regular sets: union, intersection, complement, convex hull, image, and pre‐image under suitable classes of functions. It turns out that only few of these notions are suitable in the sense of

Abstract In [6] Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant e equipped with a (certain) Hasse derivation. We propose a variant of their theory, using a sequence of e commuting Hasse derivations. In contrast to [6] our Hasse derivations are iterative.

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For the computability of subsets of real numbers, several reasonable notions have been suggested in the literature. We compare these notions in a systematic way by relating them to pairs of ‘basic’ ones. They turn out to coincide for full-dimensional convex sets; but on the more general class of regular sets, they reveal rather interesting ‘weaker/stronger’ relations. This is in contrast to single real numbers and vectors where all ‘reasonable’ notions coincide.

Abstract The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe have asserted the existence of non‐empty co‐r. e. closed sets devoid of computable points: sets which are even “large” in the sense of positive Lebesgue measure. This leads us to investigate for various classes of computable real subsets whether they always contain a (not necessarily effectively findable) computable point. (© 2008 WILEY‐VCH Verlag

I. The language L. Let L2 be the 2-sorted first order language appropriate for structures (?I, a, e), where 21 is a L-structure and a is a set of subsets of A. We call (?I, a) topological if a is a topology. We call a formula of L2 topological if it is built up using the set quantifier 3X only in the form lX(t E X A0), X does not occur positively in 0. (X occurs positively in 0 if a free occurrence of Xin 0 isIinside the scope of an even number of negation symbols. Note. Primitive symbols are A,