**
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