Computability Seminar

Finite Ramsey Theorem states that fixed $n,m,k \in \mathbb N$, there exists $N \in \mathbb N$ such that for each coloring of $[N]^n$ with $k$ colors, there is a homogeneous subset $H$ of $N$ of cardinality at least $m$. Starting with the celebrated Paris-Harrington theorem, many Ramsey-like results have been studied using different largeness notions rather than the cardinality. I will introduce the largeness notion defined by Ketonen and Solovay based on fundamental sequences of ordinals. Then I will describe an alternative and more flexible largeness notion using blocks and barriers. If time allows, I will talk about how the latter can be used to study a more general Ramsey-like result.