A thin near hexagon with 50 points

We show the existence and uniqueness of a thin near hexagon which has 50 points and an affine plane of order 3 as a local space. r 2003 Elsevier Science (USA). All rights reserved. 1. Definitions and motivation In this paper we solve the open problem mentioned at the end of [BDB]: Does there exist a near hexagon which satisfies conditions (C1), (C2) and (C3) below? ðC1Þ Every line is incident with exactly two points. ðC2Þ Every two points at distance 2 are contained in a good quad. ðC3Þ At least one local space is an affine plane of order 3. We will prove that such a near hexagon exists and that it is unique. This completes the classification of all near hexagons with good quads and with at least one local space isomorphic to an affine plane, see [BDB]. Since the paper [BDB] is published in this issue of JCT-A, we will use it as a reference for some basic notions like ‘‘near hexagon’’, ‘‘quad’’, and so on. A quad Q is called good if every point of Q is incident with the same number of lines of Q; this number then is denoted by tQ þ 1: From now on we assume that S is a near hexagon which satisfies (C1)–(C3). Since S has only lines of size 2, it can be regarded as a bipartite graph of diameter 3 (see [BDB, Lemma 1]). The edges then play the role of the lines. For a set A of vertices, let A> E-mail address: [email protected] (B. De Bruyn). 0097-3165/03/$ see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0097-3165(03)00029-3 denote the set of all vertices adjacent to every element of A: If x and y are two points at distance 2 then Qðx; yÞ :1⁄4 fx; yg,ðfx; ygÞ is the unique quad through x and y: The fact that this quad is good means that jfx; ygj 1⁄4 jfx; ygj: For every point x; we can define the following linear space LðxÞ: the points of LðxÞ are the elements of x> and the lines are all the sets fa; bg where a and b are two different vertices of x>: The linear space LðxÞ is called the local space at x: In [BDB] another definition was given, but in this special case both definitions are easily seen to be equivalent. By Theorem 5 of [BDB], we know that S has precisely v 1⁄4 50 points, that every point is incident with exactly t þ 1 1⁄4 9 lines and that every quad Q has order ð1; tQÞ with tQAf1; 2; 3g: Moreover, there exists a quad Q with tQ 1⁄4 3: The point set P of S can be partitioned into two subsets Pþ and P ; each of size 25, such that two points of the same subset are never collinear. We sometimes call a point of Pe a point of type e. The set Pe; eAfþ; g; can be given the structure of a linear space Se by taking all sets fx; yg; x; yAPe and xay; as lines. Clearly LðaÞ is a subgeometry of Se for every aAP e: 2. The structure of S Let nþ denote a point whose local space is an affine plane of order 3. We may suppose that nþAPþ: For every point x and every iAf1; 2; 3g; let NiðxÞ or shortly Ni denote the number of quads of order ð1; iÞ through x; or equivalently, the number of lines of size i þ 1 in LðxÞ: Since t þ 1 1⁄4 9; N3Af0; 1; 2; 3g: Every two lines through x are contained in a unique quad; hence 72 1⁄4 ðt þ 1Þt 1⁄4 12N3 þ 6N2 þ 2N1: Since there are jPþj 1 1⁄4 jP j 1 1⁄4 24 points at distance 2 from x; 24 1⁄4 3N3 þ 2N2 þ N1: These equations give Lemma 2.1. For any point x one of the following holds: (n) N3 1⁄4 0; N2 1⁄4 12; N1 1⁄4 0: x is called a nice point; (e) N3 1⁄4 1; N2 1⁄4 9; N1 1⁄4 3; x is called an elusive point; (g) N3 1⁄4 2; N2 1⁄4 6; N1 1⁄4 6: x is called a good point; (b) N3 1⁄4 3; N2 1⁄4 3; N1 1⁄4 9: x is called a bad point. The local space of a nice point is a Steiner system Sð2; 3; 9Þ and hence an affine plane of order 3. Lemma 2.2. If n is a nice point, then dðn;RÞp1 for every quad R of order ð1; 2Þ or ð1; 3Þ: Proof. Suppose that dðn;RÞ 1⁄4 2: If R has order ð1; 3Þ; then the four quads through n meeting R contain 5 points of G2ðRÞ; contradicting jG2ðRÞj 1⁄4 jPj jRj jG1ðRÞj 1⁄4 50 8 40 1⁄4 2: Suppose therefore that R has order ð1; 2Þ: Let Si; iAf1; 2; 3g; denote the three quads through n meeting R; put fvig :1⁄4 Si-R; and let wi denote the unique R. Blok et al. / Journal of Combinatorial Theory, Series A 102 (2003) 293–308 294