# Menachem Kojman: The Rinot-Todorcevic unified rectangles coloring theorem

Set Theory Seminar (HUJI)

We shall meet this Friday in the Hebrew University math department building, at 10 am.

Speaker: Menachem Kojman (BGU)

Title: The Rinot-Todorcevic unified rectangles coloring theorem

Abstract: During Assaf’s Purim vacation at the kindergarten, I will present
the Rinot-Todorcevic unified rectangles coloring proof and also construct
an example for non-productivity of chain conditions in BAs from $Pr_1(\lambda,\lambda,2,\omega)$.

The rectangles coloring theorem is:

For every regular cardinal $\lambda$ there is a coloring $c:[\lambda^+]^{2}\to \lambda^+$ such that for any two unbounded $A,B\subseteq \lambda^+$, for every prescribed $\gamma<\lambda^+$ there are $\alpha < \beta <\lambda^+$, $\alpha\in A$, $\beta\in B$ and $c(\alpha,\beta)=\gamma$.

The theorem is a consequence of three different results: Shelah’s proofs for $\lambda > \aleph_1$ (proved first for $\lambda > 2^{\aleph_0}$) and separately for $\lambda=\aleph_1$, and then Moore’s proof of the missing case $\lambda=\aleph_0$.

The unified proof works for all regular $\lambda$ and uses basic properties of Todorcevic walks.

The non-productivity from Pr_1 is not related directly to this coloring but to the continuation of Assaf’s proof after his return from the vacation.

See you there!