Friedman's circle theorem

Friedman's circle theorem is a theorem of Harvey Friedman that leads to an extremely fast-growing function.

Suppose you are given a fixed natural number k. Then Friedman's circle theorem says that there exists a natural number n such that, for any sequence \(C_1, C_2, \ldots, C_n\) of pairwise dijoint circles in the plane, there exist \(k \le i < j \le \lfloor n/2 \rfloor \) and a homeomorphism of the plane that maps \(C_i \bigcup C_{i+1} \bigcup \ldots \bigcup C_{2i}\) into \(C_j \bigcup C_{j+1} \bigcup \ldots \bigcup C_{2j}\).

A homeomorphism is defined as a one-to-one and onto mapping \(f\) such that both \(f\) and \(f^{-1}\) are continuous. To put it simply, both \(f\) and its inverse must be "smooth" mappings that do not take points that are close to each other to far away from each other.

Let Circle(k) be the largest n such that there exists a sequence \(C_1, C_2, \ldots, C_n\) such that no homeomorphism of the above form exists. It can be shown that Circle(1) = 5 and Circle(2) \(\ge\) 13. In general, Circle(k) grows at the rate of \(f_{\varepsilon_0}(k)\) in the FGH.