Remarks about $\gamma$-sets and Borel-dense sets

Remarks about disjoint dominating sets

We solve a number of problems posed by Hedetniemi, Hedetniemi, Laskar, Markus, and Slater concerning pairs of disjoint sets in graphs which are dominating or independent and dominating.

Two remarks about analytic sets

In this paper we give two results about analytic sets. The rst is a counterexample to a problem of Fremlin. We show that there exists ! 1 compact subsets of a Borel set with the property that no -compact subset of the Borel set covers them. In the second section we prove that for any analytic subset A of the plane either A can be covered by countably many lines or A contains a perfect subset P ...

Some remarks about Cantor sets

The classical middle-thirds Cantor set can be described as follows. Start by taking C 0 = [0, 1], the unit interval in the real line. Then put which is to say that one removes the open middle third from the unit interval to get a union of two disjoint closed intervals of length 1/3. By repeating the process one gets for each nonnegative integer j a subset C j of the unit interval which is a uni...

Borel Sets and Countable Models

We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alon...

A Course on Borel Sets