Assaf Rinot: Transforming rectangles into squares, with applications to strong colorings

Set Theory Seminar (HUJI)

Our next seminar is on this Friday, 10:00am, in Jerusalem.

Assaf Rinot continues his lecture.

Title: Transforming rectangles into squares, with applications to strong colorings

Abstract: We prove that every cardinal k which is the successor of a singular cardinal, admits a function $rts:[\kappa]^2\rightarrow[\kappa]^2$ such that every rectangle $A\times B$ is transformed by $rts$ into a square $C\times C$.
As a corollary, we get that Shelah’s notion of strong coloring $Pr_1$ coincides with the classical square bracket relation $\kappa\nrightarrow[\kappa]^2_\kappa$.

In the previous meeting, we described the history of the problem, and provided a proof to Lemma 2.4 of [1]. In the upcoming meeting, we shall provide a proof of the main result for the case of uncountable cofinality (Theorem 2.5), and outline the modifications needed for the proof of the remaining case.