Lecture I: Convex geometry and forcing

Abstract. There is a natural way to quantify the non-convexity of a subset S of a real vector space. S is close to being convex if its convexity number , the least size of a family of convex subsets of S which covers S, is small.

We are interested in uncountable convexity numbers of closed sets in R^n. In general, the following dimension conjecture is open:
In R^n there are at most n different uncountable convexity numbers of closed sets and it is consistent that there are exactly n This conjecture holds true for n=1 and n=2 and there are some hints of structure in the higher dimensions. There are connections to continuous Ramsey theory.

Prerequisites: A general idea of forcing, minimal knowledge of geometry in R^n.

Suggested reading: Convex decompositions in the plane and continuous pair colorings of the ittationals, Israel J. Math 131(2002),285-317 dvi, Convexity numbers of closed sets in R^n, Proc. AMS vol. 130 (2002) 2871-1881 dvi, More on convexity numbers of closed sets in R^n ps.

Lecture II: Continuous Ramsey theory

Abstract. The homogeneity number of a continuous coloring of the unordered pairs of a Polish spa-ce is the least number of homogeneous subsets needed to cover the whole space. Uncountable homogeneity numbers of continuous colorings turned up in planar convex geometry. Consistently they are strictly below 2^{\aleph_0}, but they cannot be more than one cardinal below the continuum. Among those colorings which have an uncountable homogeneity number, there are a minimal and a maximal coloring. It is possible to separate the homogeneity numbers of the minimal and the maximal coloring, opening up the field of investigation of what can happen in between.

Prerequisites: A general idea of forcing, maybe even of Sacks forcing and countable support iteration.

Suggested reading: Continuous Ramsey Theory on Polish spaces and covering the plane by functions dvi, and Survey of Sacks Forcing