In the early 1970’s, Hindman proved a beautiful theorem in
additive Ramsey theory asserting that for any partition of the set of
natural numbers into finitely many cells, there exists some infinite set
such that all of its finite sums belong to a single cell.
In this talk, we shall address generalizations of this statement to the
realm of the uncountable. Among other things, we shall present a
negative partition relation for the real line which simultaneously
generalizes a recent theorem of Hindman, Leader and Strauss, and a
classic theorem of Galvin and Shelah.
This is joint work with David Fernandez-Breton.