Hyper-hyper-hyper-integers

Abstract: In a conference five years ago, T. Tao reported his effort to simplify Szemer\'{e}di's original combinatorial proof of Szemer\'{e}di's theorem using nonstandard analysis. We continued his effort and presented a simple proof of the theorem for $k=4$ in CANT 2020. In this talk, we will present a simple proof of the theorem for all $k$. One of the main simplifications is that a Tower of Hanoi type induction used by Szemer\'{e}di as well as Tao is replaced by a straightforward induction. In the proof the integers with three levels of infinities are used.

number theory

Audience: researchers in the discipline

( paper )

Combinatorial and additive number theory (CANT 2022)