# Thilo Weinert: Avoiding Quadruples Using a Scale

BIU seminar in Set Theory

On 08/06/2017, 10-12, Building 604, Room 103

Speaker: Thilo Weinert (BGU)

Title: Avoiding Quadruples Using a Scale

Abstract. In 1971, Hajnal showed that the continuum hypothesis implies omega_1^2 -|-> (omega_1^2, 3)^2 and in the same year, together with Erdős, that GCH implies that for every infinite cardinal kappa and every alpha < (kappa^+)^2 we have alpha -|-> (kappa^+ * kappa, 3)^2. In the same paper they showed that for infinite cardinals kappa and alpha < (kappa^+)^2 we have (kappa^+)^2 —> (alpha, 3)^2. In 1987, together with Baumgartner, he showed that for regular kappa satisfying 2^kappa = kappa^+ = lambda we have lambda^2 -|-> (lambda * kappa, 4)^2.

In 1998, Jean Larson showed that for regular kappa and lambda = kappa^+ the existence of a scale of length lambda of functions f : kappa — > kappa implies the failure of the aforementioned partition relations shown to
be unprovable from ZFC in the seventies, i.e. lambda * kappa —> (lambda * kappa, 3)^2 and lambda^2 —> (lambda^2, 3)^2. She commented that it would be interesting to know whether this hypothesis also suffices to prove lambda^2 -|-> (lambda * kappa, 4)^2.

It does.