A short proof that N3 is not a circle containment order

AbstractA partially ordered set P is called a circle containment order provided one can assign to each x∈P a circle Cxso that $$x \leqslant y \Leftrightarrow C_x \subseteq C_y $$ . We show that the infinite three-dimensional poset N3 is not a circle containment order and note that it is still unknown whether or not [n]3 is such an order for arbitrarily large n.