Sets of type \((q+2,n)\) in \(\mathrm{PG}(3, q)\). (English) Zbl 1525.51006
A set of points of type \((n,m)\), \(n>m,\) in the projective space \(\mathrm{PG}(N,q)\) is a set \(S\) of points such that each hyperplane meets \(S\) in either \(n\) or \(m\) points, and both intersection numbers occur. In [Atti Accad. Peloritana Pericol. 96, Suppl. 2, Paper No. A7, 12 p. (2018; doi:10.1478/AAPP.96S2A7)], S. Innamorati and F. Zuanni proved that sets of type \((n,q)\) and of type \((n,q+1)\) in the three dimensional projective space \(\mathrm{PG}(3,q)\), are such that \(n=m+q\). This was conjectured in [N. Durante et al., Ric. Mat. 65, No. 1, 65–70 (2016; Zbl 1348.51003); J. Geom. 107, No. 1, 9–18 (2016; Zbl 1338.51013)]. In the paper under review, the authors prove that the same holds true for sets of points of \(\mathrm{PG}(3,q)\) of type \((n,q+2)\), namely for these sets, \(n=2q+2\). The proof relies on a result by M. Tallini Scafati [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 60, 782–788 (1976; Zbl 0376.50011)] and on counting techniques.
Reviewer: Stefano Lia (Dublin)
MSC:
| 51E20 | Combinatorial structures in finite projective spaces |
| 51E21 | Blocking sets, ovals, \(k\)-arcs |
References:
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| [4] | Durante, N., Napolitano, V., Olanda, D.: Sets of type \((q+1, n)\) in \(PG(3, q)\). J. Geom. 107, 9-18 (2016). doi:10.1007/s00022-015-0271-5 · Zbl 1338.51013 |
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