A Bose-Burton theorem for elliptic polar spaces
In [J. Comb. Theory 1, 96-104 (1966; Zbl 0152.18106)] \textit{R. C. Bose} and \textit{R. C. Burton} proved that in a finite projective space \(\text{ PG}(d,q)\) a set \(B\) of points that meets every subspace of a fixed dimension \(c\leq d\) has at least \((q^{d+1-c}- 1)/(q-1)\) points with equality if and only if \(B\) consist of the points of a subspace of dimension \(d-c\). There are various interesting generalizations and variations of this problem. The main result of the paper under review for example states that for \(0\leq c \leq n-1\) the smallest set \(B\) of points in the elliptic quadric \(Q := Q^-(2n+1,q)\) that meets every singular subspace of dimension \(c\) of \(Q\) has cardinality \((q^{n+1} + q^c)(q^{n-c} - 1)/(q-1)\). Equality holds if and only if there is a subspace \(U\) of dimension \(c-1\) of \(Q\) such that \(B = (U^{\perp}\setminus U) \cap Q\), where \(\perp\) is the polarity associated with the quadric.
scientific article; zbMATH DE number 1392715