\(m\)-clouds in generalized hexagons (Q1849873)

An \(m\)-cloud of a generalized hexagon of order \((s, t)\) is a set \(S\) of points at mutual distance 4 such that for any two points \(x\) and \(y\) of \(S\) the unique point \(z\) at distance 2 of \(x\) and \(y\) is collinear with exactly \(m+1\) points of \(S\). For an \(m\)-cloud \(S\) denote by \(S^*\) the set of all points \(z\) such that \(z\) is at distance 2 of two (different) points of \(S\). The authors obtain among others the following results on \(m\)-clouds in generalized hexagons: 1. The points of an \(m\)-cloud \(S\) are collinear with a constant number \(f+1\) points of \(S^*\). This number \(f\) is called the index of \(S\). 2. If \(S\) is an \(m\)-cloud of index \(m\), then \((S, S^*, \sim)\) is a projective plane. 3. For \(k > t-\sqrt{t}+1\), a \((k-1)\)-cloud \(S\) of index \(k\) is extendable to a \(k\)-cloud of index \(k\). 4. Assume that the generalized hexagon admits a distance-2-regular point \(p\). Let \(q\) be a point opposite to \(p\) and suppose that \(S\) is a subset of the projective plane \(\Gamma_{\pi} = (\Gamma^{+}(p,q), \Gamma^{-}(p,q), \sim)\) such that all lines of \(\Gamma_{\pi}\) intersect \(S\) in 0, 1 or \(m+1\) points. Then \(S\) is an \(m\)-cloud of \(\Gamma\). (For a point \(q\) opposite to \(p\), the set \(\Gamma^{+}(p,q)\) consists of all points of the thin subhexagon through \(p\) and \(q\) at distance 0 or 4 of \(p\). \(\Gamma^{-}(p,q)\) is the complementary point set in this thin hexagon.) 5. In an antiregular hexagon an \(m\)-cloud \(S\) with \(m \geq 2\) always fulfils \(|S^*|= 1\). The paper ends with some results on so-called dense clouds in generalized quadrangles.