Activities This Week

Let $F = (F_1, \ldots, F_m)$ be a collection of (not neccessarily distinct) sets. A (partial) rainbow set for $F$ is a set of the form $R = {x_{i_1}, \ldots, x_{i_k}}$ of distinct elements, where $1 \leq i_1 < \cdots < i_k \leq m$ and $x_{i_j}$ is an element of $F_{i_j}$. We are interested in the following question: given sufficiently many independent sets of size $a$ in a graph belonging to a certain class, there exists a rainbow independent set of size $b$. In this talk, I will present our recent results on this question, mainly regarding $H$-(induced) free graphs and graphs of bounded maximum degree. This is joint work with Ron Aharoni, Joseph Briggs and Jinha Kim.

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 others, we shall present a new theorem concerning 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.