Lorenz Halbeisen

The general counterfeit coin problem

Abstract Given c nickels among which there may be a counterfeit coin, which can only be told apart by its weight being different from the others, and moreover b balances, what is the minimal number of weighings to decide whether there is a counterfeit nickel, if so which one it is and whether it is heavier or lighter than a genuine nickel. We give an answer to this question for sequential and nonsequential strategies and we will consider the problem of more than one counterfeit coin.

Silver measurability and its relation to other regularity properties

For a ⊆ b ⊆ ω with b\a infinite, the set D = {x ∈ [ω] : a ⊆ x ⊆ b} is called a doughnut. Doughnuts are equivalent to conditions of Silver forcing, and so, a set S ⊆ [ω] is called Silver measurable, also known as completely doughnut, if for every doughnut D there is a doughnut D′ ⊆ D which is contained or disjoint from S. In this paper, we investigate the Silver measurability of ∆2 and Σ 1 2 sets of reals and compare it to other regularity properties like the Baire and the Ramsey property and Miller and Sacks measurability.

Reconstruction of Weighted Graphs by their Spectrum

It will be shown that for almost all weights one can reconstruct a weighted graph from its spectrum. This result is the opposite to the well-known theorem of Botti and Merris which states that reconstruction of non-weighted graphs is, in general, impossible since almost all (non-weighted) trees share their spectrum with another non-isomorphic tree.

RELATIONS BETWEEN SOME CARDINALS IN THE ABSENCE OF THE AXIOM OF CHOICE

2 Abstract. If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary inflnite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in speciflc permutation models and give some results provable without using the axiom of choice.

On asymptotic models in Banach Spaces

AbstractA well known application of Ramsey’s Theorem to Banach Space Theory is the notion of a spreading model

On Shattering, Splitting and Reaping Partitions

(\tilde e_i )

Symmetries between two Ramsey properties

of a normalized basic sequence (xi) in a Banach spaceX. We show how to generalize the construction to define a new creature (ei), which we call an asymptotic model ofX. Every spreading model ofX is an asymptotic model ofX and in most settings, such as ifX is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorem—a strengthened form of Ramsey’s Theorem—to generate asymptotic models with a stronger form of convergence.

A Simple Proof of Poncelet's Theorem (on the Occasion of Its Bicentennial)

Abstract A topological space X is continuously Urysohn if for each pair of distinct points x,y∈X there is a continuous real-valued function fx,y∈C(X) such that fx,y(x)≠fx,y(y) and the correspondence (x,y)→fx,y is a continuous function from X2⧹Δ to C(X), where C(X) carries the topology of uniform convergence and Δ={(x,x): x∈X} . Metric spaces are examples of continuously Urysohn spaces with the additional property that the functions fx,y depend on just one parameter. We show that spaces with this property are precisely the spaces that have a weaker metric topology. However, to find an example of a continuously Urysohn space where the functions fx,y cannot be chosen independently of one of their parameters, it is easier to consider a much simpler property than “continuously Urysohn”, given by the following definition: A topological space X is strongly separating if for each point x∈X there is a continuous, real-valued function gx such that for any z∈X, gx(x)=gx(z) implies x=z. We show that a continuously Urysohn space may fail to be strongly separating. In particular, the example that we present is a continuously Urysohn space, where the Urysohn functions fx,y cannot be chosen independently of y. This answers a question raised by David Lutzer.