SCHWEITZER 1983 Problem 4

For which cardinalities $\kappa$ do antimetric spaces of cardinality $\kappa$ exist? $(X,\varrho)$ is called an $\textit{antimetric space}$ if $X$ is a nonempty set, $\varrho : X^2 \rightarrow [0,\infty)$ is a symmetric map, $\varrho(x,y)=0$ holds iff $x=y$ , and for any three-element subset $\{a_1,a_2,a_3 \}$ of $X$ $$\varrho(a_{1f},a_{2f})+\varrho(a_{2f},a_{3f}) < \varrho(a_{1f},a_{3f})$$ holds for some permutation $f$ of $\{1,2,3 \}$ .

V. Totik