Annals of Combinatorics

, Volume 23, Issue 2, pp 423–424

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

Correction

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

In Section 4.2 of , 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  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 .

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 .

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  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 $${\mathrm {d}}(y,x_1)={\mathrm {d}}(y,x_2)={\mathrm {d}}(y,x_3)$$.

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) 