A Resolution of the de Bruijn--Erdős Consecutive-Gap Problem

Let $(x_n)_{n\geq1}$ be a sequence of distinct points on the unit circle. An $r$-span is the total length of $r$ consecutive gaps determined by the inserted points. Write $M_n^{(r)}$ and $m_n^{(r)}$ for the largest and smallest $r$-spans after the first $n$ insertions. We prove that there is an abso