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A partial classification of such dense near hexagons was obtained in [5]. In [5], ten examples were described along with four open cases. In [9], the valuations of the dense near hexagons with four points per line were studied, and these results were subsequently used in [13] to obtain the above-mentioned classification for the octagons. 7], one of the four open cases in the classification of the near hexagons was killed. In view of this, there remain now three open cases in the classification of the hexagons and also three open cases in the classification of the octagons.

D}, such that for every two points x and y at distance i from each other there are ti + 1 lines through y containing a (necessarily unique) point at distance i − 1 from x. If this is the case, then t0 = −1, t1 = 0 and td = t. Finite thick dual polar spaces of rank d and finite generalized 2dgons are examples of regular near 2d-gons. The regular near 2d-gons are precisely those near 2d-gons whose collinearity graph is a so-called distance-regular graph [2]. A semi-valuation of a point-line geometry S = (P, L, I) is a map f : P ≤ Z satisfying the property that every line L contains a (unique) point x L∞ such that f (x) = f (x L∞ ) + 1 for every point x of L distinct from x L∞ .

This implies that { f x | x ∈ L → } is an L-set. (2) Suppose there exists a line L in S at distance Φ − 1 from L → . In view of (1), it suffices to show that the maps f x , x ∈ L → , are mutually distinct. So, consider two distinct points x1→ and x2→ of L → and let xi , i ∈ {1, 2}, denote the unique point of L nearest to xi→ . Since d(xi→ , L) ⊕ Φ − 1 and d(xi→ , y) ⇔ Φ for every point y of L, we necessary have d(xi , xi→ ) = Φ − 1. Now, x1 ∼= x2 , otherwise the point x1 = x2 would lie at distance at most Φ − 2 from a point of L → since it lies at distance Φ − 1 from the points x1→ and x2→ .