Some new upper bounds for the size of partial ovoids in slim generalized polygons and generalized hexagons of order \((s,s^3)\)
A set of mutually opposite points of a generalized polygon \(\Gamma\) is called a partial ovoid. If a partial ovoid \(\mathcal O\) of a generalized polygon \(\Gamma\) is such that every element of \(\Gamma\) is at distance at most \(n/2\) from some elements of \(\mathcal O\), then \(\mathcal O\) is called an ovoid. The authors prove the following result. Let \(\Gamma\) be a generalized polygon containing a partial ovoid \(\mathcal O\). Suppose \(\Gamma\) is one of the following : an orthogonal quadrangle \(Q(5,2)\), a split Cayley hexagon \(H(2)\), a twisted triality hexagon \(T(2,8)\), or the Ree-Tits Octagon \(O(2)\). Then the number \(k\) of points of \(\mathcal O\) satisfies: (i) \(k \leq 7\) if \(\Gamma = Q(5,2)\) or \(H(2)\); (ii) \(k \leq 25\) if \(\Gamma = T(2,8)\); (iii) \(k \leq 27\) if \(\Gamma = O(2)\). Moreover, the authors prove that if \(\Gamma\) is a generalized hexagon of order \((s,s^3)\), for some integer \(s > 1\), then a partial ovoid has at most \(s^5 - s^3 + s -1\) points.
scientific article; zbMATH DE number 1553306