I have worked in the field of Euclidean Ramsey Theory. Recently I have been spending a lot of time on various polymath projects so a lot of the content here has been devoted to that. I also have spent some time on Math Overflow.
Euclidean Ramsey theory involves Ramsey theory and geometry. Let me give two sample problems. One is if the plane is two colored and we have an isoceles triangle is there a monochromatic copy of that triangle?
The other is if we have set K that lies on a sphere for any number of colors r is there a space of dimension n such that for any partition of that space into r colors there is a monochromatic set congruent to K?
These two problems are open. They are examples of the type of problems that come up in this field.