Uncountably many homogeneous real trees with the same valence

Abstract

For any cardinal $\hat{I}º\ge 2$, there is a unique complete real tree whose points all have valence $\hat{I}º$. In this note, we show that, when $\hat{I}º\ge 3$, it is necessary to assume completeness. More precisely, we show that there exist uncountably many homogeneous incomplete real trees whose points all have valence $\hat{I}º$.

8 pages