at nuclear matter saturation density, $\rho_0 = 0.16$ fm$^{-3}$. This constraint will be satisfied by any of the calculations to be described below.

Fig.\[fig:ULambda1\] shows instructive examples of the calculated\
$U_\Lambda(p_1=0;\rho)$ in nuclear matter at low densities. Notably, the result using only the two-body chiral NLO potential would actually produce too much attraction in the $\Lambda$ potential, causing an overbinding of hypernuclei. This calls for repulsive effects beyond two-body interactions, with three-body forces being the natural extension. The three-body effects are moderate (a few MeV) at densities $\rho\simeq\rho_0$, but their relative importance grows continuously as the density increases. At $\rho \simeq 2\,\rho_0$ the repulsive $\Lambda NN$ 3BF is essential in order to turn the sign of $U_\Lambda$ from attractive to repulsive. The result in fig.\[fig:ULambda1\] is produced using a combination of three-body constants, $(H_1,H_2)$, that evidently satifies the constraint eq.(\[eq:USNM\]). However, this choice of $H_1$ and $H_2$ is not uniquely determined. It turns out that any pair $(H_1,H_2)$ that lies on either of the two solid curves in fig.\[fig:H1H2\] is compatible with the condition, $U_\Lambda(\rho=\rho_0) = - 30$MeV. On these two constrained curves one can identify points $(H_1,H_2)$ for which the $\Lambda$ single-particle potential at a higher density, $U_\Lambda(\rho=3\rho_0)$, becomes 