On c+-chromatic graphs with small bounded subgraphs

It is shown that if ZF is consistent then so is ZFC+2 is as large as you wish + there is a graph with cardinality and chromatic number (2) + such that every subgraph of cardinality 2 has chromatic number .The preparation of this paper was supported by Hungarian National Foundation for Scientific Research (OTKA), grant no. 1805.     

A note on a set-mapping problem of Hajnal and Máté

It is consistent that there exists a set mappingF with <F(, )< for + 2 >w ₂ with no uncountable free sets.Research supported by Hungarian National Research Fund No. 1805 and 1908.     

Sacks forcing and the total failure of Martin's Axiom

Using side-by-side Sacks forcing, it is proved relatively consistent that the continuum is large and Martin's Axiom fails totally, that is, every c.c.c. space is the union of ?₁ nowhere dense sets (equivelently, if P is a nontrivial partial ordering with the countable chain condition, then there are ?₁ dense sets in P such that no filter in P meets them all).     

Non-baire unions in category bases

We consider a partition of a spaceX consisting of a meager subset ofX and obtain a sufficient condition for the existence of a subfamily of this partition which gives a non-Baire subset ofX. The condition is formulated in terms of the theory of J. Morgan [1].     

What must and what need not be contained in a graph of uncountable chromatic number?

We investigate the following problem: What countable graphs must a graph of uncountable chromatic number contain? We define two graphsΓ andΔ which are very similar and we show thatΓ is contained in every graph of uncountable chromatic number, whileΔ is (consistently) not.     

On the constructive Dedekind reals

In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.     

Non-nesting actions of Polish groups on real trees

We prove that if a Polish group G with a comeagre conjugacy class has a non-nesting action on an R-tree, then every element of G fixes a point.     

A strongly non-Ramsey order type

It is consistent that there is an order type for which holds for every type .Research partially supported by Hungarian National Science Grant OTKA 016391.Partially supported by the European Communities (Cooperation in Science and Technology with Central and Eastern European Countries) contract number ERBCIPACT930113.     

A Characterization of Generalized Příkrý Sequences

A generalization of Příkry's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkry generic sequences reminiscent of Mathias' criterion for Příkry genericity is provided, together with a maximality theorem which states that a generalized Příkry sequence almost contains every other one lying in the same extension. This forcing can be used to falsify the covering lemma for a higher core model if there is an inner model with infinitely many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ. During the research for this paper the author was supported by DFG-Project Je209/1-2.     

The Square of Opposition and the Four Fundamental Choices

Each predicate of the Aristotelian square of opposition includes the word “is”. Through a twofold interpretation of this word the square includes both classical logic and non-classical logic. All theses embodied by the square of opposition are preserved by the new interpretation, except for contradictories, which are substituted by incommensurabilities. Indeed, the new interpretation of the square of opposition concerns the relationships among entire theories, each represented by means of a characteristic predicate. A generalization of the square of opposition is achieved by not adjoining, according to two Leibniz’ suggestions about human mind, one more choice about the kind of infinity; i.e., a choice which was unknown by Greek’s culture, but which played a decisive role for the birth and then the development of modern science. This essential innovation of modern scientific culture explains why in modern times the Aristotelian square of opposition was disregarded. This work was completed with the support of our -pert.     

Random relaxed controls and partially observed stochastic systems

We introduce a class of generalized controls called random relaxed controls, and show that under quite general conditions, a partially observed, controlled diffusion will have an optimal random relaxed control whose cost equals the infimum over the costs of all ordinary controls. We also show that the optimal admissible control can be approximated arbitrarily well by very simple, ordinary controls. The proofs are based on a close analysis of the standard parts of nonstandard controls.     

Cofinal types of ultrafilters

We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ω₁]^<ω. We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.     

Amorphe Potenzen kompakter Räume

A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT ₂ space is compact. TA2: An amorphous power of a compactT ₂ space which as a set is wellorderable is compact. In ZF⁰TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF⁰RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF⁰. But TA2 cannot be proved in ZF⁰ alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD ₃ (a set is wellorderable, if each of its infinite subsets is structured) together with RT.

Diese Arbeit ist Teil der Habilitationsschrift des Verfassers im Fachgebiet Mathematische Analysis an der Technischen Universität Wien.     

Multiplicity of solutions for gradient systems with strong resonance at higher eigenvalues

In this paper we establish existence and multiplicity of solutions for an elliptic system which has strong resonance at higher eigenvalues. We describe the resonance using an eigenvalue problem with indefinite weights. In all results we use Variational Methods and the Morse theory.     

A Geometric Construction for Some Ovoids of the Hermitian Surface

Multiple derivation of the classical ovoid of the Hermitian surface of is a well known, powerful method for constructing large families of non classical ovoids of . In this paper, we shall provide a geometric costruction of a family of ovoids amenable to multiple derivation.     

Arcs in the plane

Assuming PFA, every uncountable subset E of the plane meets some C¹ arc in an uncountable set. This is not provable from MA(ℵ₁), although in the case that E is analytic, this is a ZFC result. The result is false in ZFC for C² arcs, and the counter-example is a perfect set.