# Largeness notions

## Andrea Volpi

This is joint work with Alberto Marcone and Antonio Montalb'an. 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 obtained by replacing cardinality with different largeness notions have been studied. 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. Finally, I will talk about how the latter can be used to study a more general Ramsey-like result.