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We study systems of orientations on triples that satisfy the following so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ implies $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O; we denote a P3O realizable by points as p-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-T3O, is the order type of a point set in general position. In our paper "Orientation of convex sets" we defined a 3-order on pairwise intersecting convex sets; such a P3O is called a C-P3O. In this paper we extend this 3-order to pairwise intersecting good covers; such a P3O is called a GC-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a C-T3O and a GC-T3O, respectively. The main result of this paper is that there is a p-T3O that is not a GC-T3O, implying also that it is not a C-T3O -- this latter problem was left open in our earlier paper. Our proof involves several combinatorial and geometric observations that can be of independent interest. Along the way, we define several further special families of GC-T3O's.

We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture., LIPIcs, Vol. 224, 38th International Symposium on Computational Geometry (SoCG 2022), pages 32:1-32:13