A characterization of the natural embedding of the split Cayley hexagon in \(\mathrm{PG}(6,q)\) by intersection numbers in finite projective spaces of arbitrary dimension. (English) Zbl 1277.05028
Summary: We prove that a non-empty set \(\mathcal{L}\) of at most \(q^5+q^4+q^3+q^2+q+1\) lines of \(\mathrm{PG}(n,q)\) with the properties that (1) every point of \(\mathrm{PG}(n,q)\) is incident with either \(0\) or \(q+1\) elements of \(\mathcal{L}\), (2) every plane of \(\mathrm{PG}(n,q)\) is incident with either \(0\), \(1\) or \(q+1\) elements of \(\mathcal{L}\), (3) every solid of \(\mathrm{PG}(n,q)\) is incident with either \(0\), \(1\), \(q+1\) or \(2_q+1\) elements of \(\mathcal{L}\), and (4) every four-dimensional subspace of \(\mathrm{PG}(n,q)\) is incident with at most \(q^3-q^2+4q\) elements of \(\mathcal{L}\) is necessarily the set of lines of a split Cayley hexagon \(H(q)\) naturally embedded in \(\mathrm{PG}(6,q)\).
MSC:
| 05B25 | Combinatorial aspects of finite geometries |
References:
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