Quasimetric spaces with few lines

Chen and Chv\'atal conjectured in 2008 that in any finite metric space either there is a line containing all the points - a universal line -, or the number of lines is at least the number of points. This is a generalization of a classical result due to Erd\H{o}s that says that a set of $n$ non-collinear points in the Euclidean plane defines at least $n$ different lines. A line of a metric space with metric $\rho$ is defined in terms of a notion called the betweenness of the space which is the set of all triples $(x,z,y)$ such that $\rho(x,y)=\rho(x,z)+\rho(z,y)$. In this work we prove that for each $n\geq 4$ there are $p_3(n)$ non isomorphic betweennesses arising from \emph{quasimetric} spaces with $n$ points, without universal lines and with exactly 3 lines, where $p_3(n)$ is the number of partitions of an integer $n$ into three parts. We also prove that for $n\geq 5$, there are $2p_3(n-1)$ non isomorphic betweennesses arising from quasimetric spaces on $n$ points, without universal lines and with exactly 4 lines. Here two betweennesses are isomorphic if they are isomorphic as relational structures. None of the betweennesses mentioned above is metric which implies that Chen and Chv\'atal's conjecture is valid for metric spaces with at most five points.