Advertisement

Annals of Combinatorics

, Volume 23, Issue 2, pp 423–424 | Cite as

Correction to: On Semi-finite Hexagons of Order (2, t) Containing a Subhexagon

  • Anurag Bishnoi
  • Bart De BruynEmail author
Correction
  • 72 Downloads

1 Correction to: Ann. Comb. 20 (2016) 433–452  https://doi.org/10.1007/s00026-016-0315-z

In Section 4.2 of [1], we showed that there does not exist any infinite near hexagon \({\mathcal {N}}\) of order (2, t) that contains an isometrically embedded subgeometry \({\mathcal {H}}\) isomorphic to H(2). The proofs of Lemmas 4.6 and 4.7 in [1] have been spoiled by the same error: points of \({\mathcal {N}}\) at distance 1 from \({\mathcal {H}}\) are not necessarily collinear with a unique point of \({\mathcal {H}}\) (see Page 446, Line −8 and Page 447, Line 2). This is true in case \({\mathcal {N}}\) is a generalised hexagon, but not if \({\mathcal {N}}\) is a general near hexagon. Luckily, these errors can be corrected.

The following proof should replace the proof of Lemma 4.6 in [1].

Lemma 1

There are only finitely many points of type \(B_1\) in \({\mathcal {N}}\).

Proof

Let \(\mathcal {B}\) denote the set of those points of \({\mathcal {N}}\) that have type \(B_i\) for some \(i \in \{ 2,3,4,5 \}\). Then \(\mathcal {B}\) is finite by [1, Lemma 4.5]. Let \(\mathcal {A}\) denote the set of those points of \({\mathcal {N}}\) that have type A, i.e., the points of \({\mathcal {H}}\). Then the set \(\mathcal {A} \cup \mathcal {B}\) is also finite. Let x be a point of type \(B_1\) in \({\mathcal {N}}\). Then by [1, Lemma 4.2], x is at distance 1 from \({\mathcal {H}}\), and since \({\mathcal {O}}_{f_x}\) is a singleton, there exists a unique point \(\pi (x)\) in \({\mathcal {H}}\) collinear with x. If x is only collinear with points of type A, \(B_1\) or C, then by the same reasoning as in the proof of [1, Theorem 4.4], we get a contradiction. So, x is collinear with at least one point of \(\mathcal {B}\), and we have already seen that it is collinear with at least one point of \(\mathcal {A}\). Thus x is the common neighbour of two points at distance 2 in the finite set \(\mathcal {A} \cup \mathcal {B}\). Since each such pair of points at distance 2 in the near polygon \({\mathcal {N}}\) has finitely many (at most five) common neighbours, we see that the set of points of type \(B_1\) must be finite; in fact, the cardinality of this set is bounded by five times the number of unordered pairs of points at distance 2 in \(\mathcal {A} \cup \mathcal {B}\). \(\square \)

The following proof should replace the proof of Lemma 4.7 in [1].

Lemma 2

There are only finitely many points of type C in \({\mathcal {N}}\).

Proof

Let x be a point of type C in \({\mathcal {N}}\). Then the set of points of \({\mathcal {H}}\) at distance 2 from x is a 1-ovoid of \({\mathcal {H}}\) and hence it has cardinality 21. Let \(S_x\) be the set of common neighbours between x and the elements of \({\mathcal {O}}_{f_x}\) (the 1-ovoid of \({\mathcal {H}}\) induced by x). By [1, Lemma 4.2], each element y of \(S_x\) has type \(B_i\) for some \(i \in \{ 1,2,\ldots ,5 \}\) and hence by [1, Table 3] y is collinear with at most nine points of \({\mathcal {H}}\). Therefore, \(|S_x| \ge \frac{21}{9}\), and we get two points of the set \(\Gamma _1({\mathcal {H}})\) at distance 2 from each other having x as a common neighbour. By [1, Lemma 4.5] and Lemma 1, the set \(\Gamma _1({\mathcal {H}})\) is finite. A similar reasoning as in the proof of Lemma 1 then shows that there are only finitely many points of type C in \({\mathcal {N}}\). \(\square \)

The rest of the discussion in Section 4.2 of [1] can remain as it is. In the proof of Lemma 4.3, there is however a typo. The condition \({\mathrm {d}}(x,y_1)={\mathrm {d}}(x,y_2)={\mathrm {d}}(x,y_3)\) should be replaced with \({\mathrm {d}}(y,x_1)={\mathrm {d}}(y,x_2)={\mathrm {d}}(y,x_3)\).

Notes

Reference

  1. 1.
    Bishnoi, A., De Bruyn, B.: On semi-finite hexagons of order \((2,t)\) containing a subhexagon. Ann. Comb. 20(3), 433–452 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGentBelgium

Personalised recommendations