Wednesday, March 2 from 3 to 4pm

*Room*: MB 124

*Speaker*: Shehzad Ahmed (Ohio University)

*Title*: Jónsson successors of singular cardinals

*Abstract*: We say that a cardinal $\lambda$ is a Jónsson cardinal if it satisfies the following weak Ramsey-type property: given any coloring $F:[\lambda]^{<\omega}\to \lambda$ of the finite subsets of $\lambda$ in $\lambda$-many colors, there exists a set $H\in[\lambda]^\lambda$ such that the range of $F\upharpoonright [H]^{<\omega}$ is a proper subset of $\lambda$. One of the big open questions in combinatorial set theory is whether or not the existence of a singular cardinal $\mu$ such that $\mu^+$ is a Jónsson cardinal is even consistent. The goal of this talk is to explain why this problem has proven so difficult, and to (time permitting) briefly survey ongoing research in the area.