The collinearity graph of the \(O^-(8,2)\) quadric is not geometrisable
The author proves a conjecture of the reviewer and \textit{V. D. Tonchev} [Sankhyā, Ser. A 54, Spec. Vol., 137-145 (1992; Zbl 0883.51003)], that the collinearity graph of the elliptic quadric in \(\text{PG}(7,2)\) is not geometric. By considering the three hyperplanes through a \(\text{PG}(5,2)\) that intersects the quadric in a hyperbolic quadric, the graph can be split in 4 disjoint subgraphs, one of them being the line graph of \(\text{PG}(3,2)\), while the other three graphs are the complements of the triangular graph T(8). The author uses this fact to give a nice combinatorial and computer-free proof of the conjecture.
scientific article; zbMATH DE number 1115734