Dual blocking sets in projective and affine planes
Let \({\mathcal F}\) be the set of lines of a projective or affine plane. A blocking set for \({\mathcal F}\) is a set which meets every member of \({\mathcal F}\) but contains none. A dual blocking set for \({\mathcal F}\) is a set which meets every blocking set for \({\mathcal F}\) but contains no member of \({\mathcal F}\). A line oval in a projective plane of order n is a set of \(n+2\) lines, no three concurrent. A line oval in an affine plane of order n is a line oval in the corresponding projective plane, one of whose lines is the line at infinity; in other words, a set of \(n+1\) lines, no three concurrent and no two parallel. The authors prove the following theorems. Theorem 1. Let S be a dual blocking set in a projective plane of order \(n\geq 3\). Then \(| S| \leq n(n+1)/2\). Equality holds if and only if either (a) \(S=\cup {\mathcal L}\setminus L\), where \({\mathcal L}\) is a line oval and \(L\in {\mathcal L}\); or (b \(n=3\) and S is the complement of the union of two lines. Theorem 2. Let S be a dual blocking set in an affine plane of order \(n\geq 4\). Then \(| S| \leq n(n-1)/2\). Equality holds if and only if \(S=\cup {\mathcal L}\setminus L\), where \({\mathcal L}\) is a line oval and \(L\in {\mathcal L}\).
projective and affine planes. dual blocking set
scientific article; zbMATH DE number 4078133