Orders With Ten Elements Are Circle Order
Roman Bayon, Nik Lygerōs, Jean‐Sébastien Sereni · 2007
With a new method based on the notion of genetic algorithm and the explicit enumeration of orders, we prove that all orders on at most 10 elements are circle orders. This theorem represents the best partial result on Sidney-Sidney-Urrutia Conjecture. 1 Order Dimension and Circle Orders We are interested in orders of small size which are circle orders, in relation to their dimension (see [6]). As is well known, the finite posets of dimension at most two are just those which have inclusion representations using closed intervals of the real line R. Because a closed interval of R can also be considered as a sphere in R1,it is natural to ask which posets have inclusion representations using circular disks in R2. For historical reasons, these posets are called circle orders. Schneinerman and Wierman [11] showed in that Z3 is not a circle order, and then Hulbert [10] showed that the same holds for N3. Urrutia, after having proved that all finite orders with dimension at most three are regular n-gon orders for all n ≥ 3, conjectured that all