Long games and ?-projective sets

Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

In a graph Γ = (V,E) a vertex v is resolved by a vertex-set S = {v1, . . . , vn} if its (ordered) distance list with respect to S, (d(v, v1), . . . , d(v, vn)), is unique. A set A ⊂ V is resolved by S if all its elements are resolved by S. S is a resolving set in Γ if it resolves V . The metric dimension of Γ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving...

Forcing axioms and projective sets of reals

This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + “every projective set of reals is Lebesgue measurable” is equiconsistent with ZFC + “th...

On Discrete Borel Spaces and Projective Sets

Let J denote the unit interval, S—IXI the unit square; Cj and Cs the class of all subsets of I and 6, respectively. By Cj X Cj is meant the or-algebra on S generated by rectangles with sides in C/. The purpose of this note is to prove the following theorem (which settles a problem of S. M. Ulam) and observe some of its consequences. Without explicit mention, the axiom of choice has been assumed...

Finite Projective Plane Geometries and Difference Sets

Pascal-Brianchon Sets in Pappian Projective Planes Pascal-Brianchon Sets in Pappian Projective Planes

It is well-known that Pascal and Brianchon theorems characterize conics in a Pappian projective plane. But, using these theorems and their modifications we shall show that the notion of a conic (or better a Pascal-Brianchon set) can be defined without any use of theory of projectivities or of polarities as usually.