On standard two-intersection sets in \(\mathrm{PG}(r,q)\). (English) Zbl 1398.51005
In \(\mathrm{PG}(r,q)\), a \(k\) set of type \((m,m + q^{{r-1}\over{2}})_{r-1}\) is a set of \(k\) points such that each subspace of dimension \(r-1\) meets the set of points in either \(m\) or \(m + q^{{r-1}\over{2}}\) points. Moreover, these are known as the standard parameters for two-intersection sets. The author proves the following:
If \(H\) and \(K\) are two standard two-intersection sets in \(\mathrm{PG}(r,q)\) such that \(H \neq K^c\) and \(H \cap K = \emptyset\), then \(H \cup K\) is a standard two-intersection set if and only if \(H\) and \(K\) have size of the same type. If \(H\) and \(K\) are two standard two-intersection sets in \(\mathrm{PG}(r,q)\) having size of the same type such that \(H \cap K \neq \emptyset\), then \(H \cap K\) is a standard two-intersection set if and only if \(H \cup K\) is a standard two-intersection set. If \(H\) and \(K\) are two standard two-intersection sets in \(\mathrm{PG}(r,q)\) having size of different type such that \(H \not \subseteq K\), \(K \not \subseteq H\), \(H \cap K \neq \emptyset\), \(|H/K| \neq \theta_r(q)/2\) and \(|K/H| \neq \theta_r(q)/2\), then if \(H \cap K\) (respectively \(H \cup K\)) is a standard two-intersection set, we have that \(H \cup K\) (respectively \(H \cap K\)) is not a standard two-intersection set.
If \(H\) and \(K\) are two standard two-intersection sets in \(\mathrm{PG}(r,q)\) such that \(H \neq K^c\) and \(H \cap K = \emptyset\), then \(H \cup K\) is a standard two-intersection set if and only if \(H\) and \(K\) have size of the same type. If \(H\) and \(K\) are two standard two-intersection sets in \(\mathrm{PG}(r,q)\) having size of the same type such that \(H \cap K \neq \emptyset\), then \(H \cap K\) is a standard two-intersection set if and only if \(H \cup K\) is a standard two-intersection set. If \(H\) and \(K\) are two standard two-intersection sets in \(\mathrm{PG}(r,q)\) having size of different type such that \(H \not \subseteq K\), \(K \not \subseteq H\), \(H \cap K \neq \emptyset\), \(|H/K| \neq \theta_r(q)/2\) and \(|K/H| \neq \theta_r(q)/2\), then if \(H \cap K\) (respectively \(H \cup K\)) is a standard two-intersection set, we have that \(H \cup K\) (respectively \(H \cap K\)) is not a standard two-intersection set.
Reviewer: Steven T. Dougherty (Scranton)
MSC:
| 51E20 | Combinatorial structures in finite projective spaces |
| 05B25 | Combinatorial aspects of finite geometries |
References:
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