Metric spaces give rise naturally to betweenness relations through the associated lens of generalised triangle (in)equalities. Examples include the usual metric betweenness of Karl Menger [1] whereby a point $c$ is said to be between points $a$ and $b$ in a metric space $(X,\rho)$ if $\rho(a,b) = \rho(a,c) + \rho(c,b)$. Another ultrametric version, contrasting sharply with Menger betweenness but aligning strongly with subcontinuum betweenness amongst hereditarily indecomposable continua, is where we declare $c$ to be between $a$ and $b$ if $\rho(a, b) = \max {\rho(a,c), \rho(c,b)}$. Such betweenness relations induced by metrics with values in a finite set turn out to be of interest through a natural correlation with simple graphs. We exploit this to identify when a given betweenness relation is graph-induced; namely, that edges between vertices (points of $X$) can be labelled from the set ${1,2}$ in such a way that the associated Menger betweenness relation from this metric (with values in the set ${0,1,2}$) coincides with the original betweenness relation.

This is joint work with Paul Bankston (Marquette University, Wisconsin) and Steve Watson, York University, Toronto.

[1] Karl Menger, Untersuchungen "{u}ber allgemeine Metrik, Math. Ann. 100 (1928), 75–163