Near polygons with a nice chain of sub-near polygons

摘要

In (Eur. J. Combin. 24 (2003) 631) we defined a class of near polygons and conjectured that the near 2n-gons from this class are precisely those near polygons which satisfy the following properties: (i) every line is incident with exactly three points, (ii) every two points at distance 2 have at least two common neighbours, (iii) there exists a chain F-0 subset of F-1 subset of (...) subset of F-n of geodetically closed sub-near polygons with the property that the sub-near 2i-gon F-i, i is an element of {0,..., n - 1}, is big in the sub-near 2(i + 1)-gon Fi+1. In the present paper we present a proof of this conjecture. (C) 2004 Elsevier Inc. All rights reserved.