The Ramsey Theory of Ordinals has been investigated over the last decades and a large variety of results have been attained. The talk is going to focus on the Ramsey Theory of finite multiples both of infinite cardinals and, in some cases products of two infinite cardinals. This leads to problems in finite combinatorics similar to the calculation of finite Ramsey numbers. On the one hand, exact results are usually only obtainable if the natural numbers involved remain somewhat small. On the other hand, sometimes asymptotic results can be attained.

More concretely, for any ordinal \(\alpha\) and \(\beta\), let \(r(\alpha, \beta)\) denote the least ordinal \(\gamma\) such that any colouring of the pairs in \(\gamma\) in black and white either allows for a homogeneously white subset of order-type \(\alpha\) or a homogeneously black subset of order-type \(\beta\). Since the nineties it is known that the growth of \(r(n, 3)\) is of order \(n^2/log(n)\). It turns out that for any infinite cardinal \(\lambda\), we have \(r(\lambda * n, 3) = \lambda * r(I_n, L_3)\) where the growth of \(r(I_n, L_3)\) is of order \(n^2/log(n)\) as well. Similarly, if \(\kappa > \lambda\) is weakly compact, we have \(r(\kappa * \lambda * n, 3) = \kappa * \lambda * r(I_n, S_3)\) where, again, the growth of \(r(I_n, L_3)\) is of order \(n^2/log(n)\). Finally there is a finitary characterisation of the Ramsey numbers \(r(\omega^2 *n, k)\) for natural numbers \(n\) and \(k\). However the growth behaviour of \(r(\omega^2 * n, 3)\) is still unknown.

This is partly joint work with Ferdinand Ihringer and Deepak Rajendraprasad.

The talk will be held in a hybrid mode via Zoom. Students at Uni Wien are strongly encouraged to attend the seminar in person.