Blocking set

In geometry, specifically projective geometry, a blocking set is a set of points in a projective plane which every line intersects and which does not contain an entire line. The concept can be generalized in several ways. Instead of talking about points and lines, one could deal with n-dimensional subspaces and m-dimensional subspaces, or even more generally, objects of type 1 and objects of type 2 when some concept of intersection makes sense for these objects. A second way to generalize would be to move into more abstract settings than projective geometry. One can define a blocking set of a hypergraph as a set that meets all edges of the hypergraph. In geometry, specifically projective geometry, a blocking set is a set of points in a projective plane which every line intersects and which does not contain an entire line. The concept can be generalized in several ways. Instead of talking about points and lines, one could deal with n-dimensional subspaces and m-dimensional subspaces, or even more generally, objects of type 1 and objects of type 2 when some concept of intersection makes sense for these objects. A second way to generalize would be to move into more abstract settings than projective geometry. One can define a blocking set of a hypergraph as a set that meets all edges of the hypergraph. In a finite projective plane π of order n, a blocking set is a set of points of π which every line intersects and which contains no line completely. Under this definition, if B is a blocking set, then complementary set of points, πB is also a blocking set. A blocking set B is minimal if the removal of any point of B leaves a set which is not a blocking set. A blocking set of smallest size is called a committee. Every committee is a minimal blocking set, but not all minimal blocking sets are committees. Blocking sets exist in all projective planes except for the smallest projective plane of order 2, the Fano plane. It is sometimes useful to drop the condition that a blocking set does not contain a line. Under this extended definition, and since, in a projective plane every pair of lines meet, every line would be a blocking set. Blocking sets which contained lines would be called trivial blocking sets. In any projective plane of order n (each line contains n + 1 points), the points on the lines forming a triangle without the vertices of the triangle (3(n - 1) points) form a minimal blocking set (if n = 2 this blocking set is trivial) which in general is not a committee. Another general construction in an arbitrary projective plane of order n is to take all except one point, say P, on a given line and then one point on each of the other lines through P, making sure that these points are not all collinear (this last condition can not be satisfied if n = 2.) This produces a minimal blocking set of size 2n. A projective triangle β of side m in PG(2,q) consists of 3(m - 1) points, m on each side of a triangle, such that the vertices A, B and C of the triangle are in β, and the following condition is satisfied: If point P on line AB and point Q on line BC are both in β, then the point of intersection of PQ and AC is in β. A projective triad δ of side m is a set of 3m - 2 points, m of which lie on each of three concurrent lines such that the point of concurrency C is in δ and the following condition is satisfied: If a point P on one of the lines and a point Q on another line are in δ, then the point of intersection of PQ with the third line is in δ. Theorem: In PG(2,q) with q odd, there exists a projective triangle of side (q + 3)/2 which is a blocking set of size 3(q + 1)/2. Theorem: In PG(2,q) with q even, there exists a projective triad of side (q + 2)/2 which is a blocking set of size (3q + 2)/2.