Fernando Mário de Oliveira Filho (CWI Amsterdam): Lower bounds for the measurable chromatic number of the Euclidean space
The chromatic number of the n-dimensional Euclidean space is the minimum
number of colours that are needed to colour the points of the space so
that no two points at distance 1 get the same colour. One speaks of the
measurable chromatic number when the colour classes, which are the sets of
points that get the same colour, are restricted to be Lebesgue measurable.
Determining the chromatic number of the plane is a famous open problem
going back to Nelson and Isbell, and Erdös and Hadwiger. It is only
known that it lies between 4 and 7. Falconer considered the measurable
chromatic number and could improve the lower bound from 4 to 5.
We will discuss this problem and show how to use semidefinite programming
together with the theory of orthogonal polynomials to obtain lower bounds
for the measurable chromatic number of R^n for high dimensions.