Two conjectures regarding dense near polygons with three points on each line
A near polygon is a partial linear space \({\mathcal S}=({\mathcal P},{\mathcal L})\) with the property that for every point \(p\in {\mathcal P}\) and for every line \(L\in {\mathcal L}\) there exists a unique point on \(L\) nearest to \(p\). Here distances \(d\) are measured in the collinearity graph. If \(n=\text{ diam}({\mathcal S})\), then \({\mathcal S}\) is called a near \(2n\)-gon. If subspace \(X\) of \({\mathcal S}\) is geodetically closed, then \({\mathcal S}_X\) is a sub near polygon of \({\mathcal S}\). A geodetically closed sub near polygon \({\mathcal S}_X\) is called big, if every point outside \({\mathcal S}_X\) is collinear with a unique point of \({\mathcal S}_X\). If a geodetically closed sub near polygon \({\mathcal S}_X\) is a nondegenerate generalized quadrangle, then \(X\) (and \({\mathcal S}_X\)) are called a quad. A near polygon is called dense if every line is incident with at least three points and if every two points at distance 2 have at least two common neighbours. There are the infinite classes of near polygons \(Q^D(2n,2)\), \(H^D(2n-1,4)\), \({\mathbf G}_n\), \({\mathbf H}_n\) and \({\mathbf I}_n\) and three sporadic near hexagons \({\mathbf E}_1\), \({\mathbf E}_2\) and \({\mathbf E}_3\). Put now \(C_2=\{Q(5,2)\}\), \(C_3=\{H^D(5,4),{\mathbf G}_3, {\mathbf E}_1\}\), \[ \begin{aligned} C_n&= \{{\mathbf G}_n,H^D(2n-1,4)\}\cup \biggl(\bigcup_{2\leq i\leq n-1} C_i\otimes C_{n+1-i}\biggr)\quad \text{for every } n\geq 4,\\ D_2&= C_2,\\ D_n&= \{{\mathbf G}_n,H^D(2n-1,4)\}\cup \biggl(\bigcup_{2\leq i\leq n-1} D_i\otimes D_{n+1-i}) \quad \text{for every } n\geq 3,\end{aligned} \] \({\mathcal C}=C_2\cup C_3\cup\dots\), and \({\mathcal D}=D_2\cup D_3\cup\dots\)\ . Define \({\mathcal M}=\{{\mathbf O},{\mathbf L}_3, {\mathbf E}_2, {\mathbf E}_3\}\cup {\mathcal C}\cup \{Q^D(2n,2)\mid n\geq 2\}\cup \{{\mathbf H}_n\mid n\geq 3\}\cup \{{\mathbf I}_n\mid n\geq 4\}\), and \({\mathcal N}=\{{\mathbf O},{\mathbf L}_3, {\mathbf E}_3\}\cup {\mathcal D}\cup \{Q^D(2n,2)\mid n\geq 2\}\cup \{{\mathbf H}_n\mid n\geq 3\}\cup \{{\mathbf I}_n\mid n\geq 4\}\). Let \({\mathcal M}^\times\), respectively \({\mathcal N}^\times\), denote the set of all near polygons obtained by taking the direct product of some members of \({\mathcal M}\), respectively \({\mathcal N}\). Conjecture 1. Every dense near polygon with three points on each line is isomorphic to one of the elements of \({\mathcal M}^\times\). Conjecture 2. Every dense near \(2n\)-gon, \(n\geq 0\), with three points on each line containing a big geodetically closed sub near \(2(n-1)\)-gon \(F\), which is isomorphic to an element of \({\mathcal N}^\times\), is itself isomorphic to an element of \({\mathcal N}^\times\). By Theorem 3 the Conjecture 2 is true for \(F\in {\mathcal N}^\times\) as soon as it is true for every \(F\in {\mathcal N}\). The Conjecture 2 is true for \(F\in {\mathcal N}-{\mathcal D}\) (Theorems 4-7).
scientific article; zbMATH DE number 1970583