On extremal solutions of martingale problems

Martingale Problems and Path Properties of Solutions

Existence and uniqueness of solutions of martingale problems, not only in the class of r.c.l.l. or continuous process, but also in the class of progressively measurable processes has become important. In this article we give several examples which show that existence or uniqueness in any one class need not imply the same in another class. We also give an example of a well-posed martingale probl...

On Multigraph Extremal Problems

1 . Introduction . In this paper we shall consider multigraphs and digraphs (= directed graphs) with bounded multiplicity : an integer r is fixed and we shall assume, that the considered multigraphs or digraphs have no loops, further, if u and v are two vertices of a multigraph M, they can be joined by more than one edge, however, they cannot be joined by more than r edges. In case of digraphs ...

Extremal problems on ∆-systems

A family of sets is called a ∆-system (respectively, a weak ∆-system) if the intersection of any two sets is the same (respectively, the cardinality of the intersection of any two sets is the same). In 1960, P.Erdős and R.Rado started studying the maximum size of a k-uniform hypergraph not containing a ∆-system of a given size. The aim of the present article is to survey the progress and state ...

Extremal solutions to some art gallery and terminal-pairability problems

The thesis consists of two parts. In both parts, the problems studied are of significant interest, but are either NP-hard or unknown to be polynomially decidable. Realistically, this forces us to relax the objective of optimality or restrict the problem. As projected by the title, the chosen tool of this thesis is an extremal type approach. The lesson drawn by the theorems proved in the thesis ...

Extremal problems on set systems