Blocking set


In geometry, specifically projective geometry, a blocking set is a set of points in a projective plane that every line intersects and that 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.

Definition

In a finite projective plane π of order n, a blocking set is a set of points of π that every line intersects and that contains no line completely. Under this definition, if B is a blocking set, then complementary set of points, π\B 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 this setting.

Examples

In any projective plane of order n, the points on the lines forming a triangle without the vertices of the triangle form a minimal blocking set 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 produces a minimal blocking set of size 2n.
A projective triangle β of side m in PG consists of 3 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 with q odd, there exists a projective triangle of side /2 which is a blocking set of size 3/2.
Theorem: In PG with q even, there exists a projective triad of side /2 which is a blocking set of size /2.
Theorem: In PG, with p a prime, there exists a projective triad of side /2 which is a blocking set of size /2.

Size

One typically searches for small blocking sets. The minimum size of a blocking set of is called.
In the Desarguesian projective plane of order q, PG, the size of a blocking set B is bounded:
When q is a square the lower bound is achieved by any Baer subplane and the upper bound comes from the complement of a Baer subplane.
A more general result can be proved,
Any blocking set in a projective plane π of order n has at least points. Moreover, if this lower bound is met, then n is necessarily a square and the blocking set consists of the points in some Baer subplane of π.
An upper bound for the size of a minimal blocking set has the same flavor,
Any minimal blocking set in a projective plane π of order n has at most points. Moreover, if this upper bound is reached, then n is necessarily a square and the blocking set consists of the points of some unital embedded in π.
When n is not a square less can be said about the smallest sized nontrivial blocking sets. One well known result due to Aart Blokhuis is:
Theorem: A nontrivial blocking set in PG, p a prime, has size at least 3/2.
In these planes a projective triangle which meets this bound exists.

History

Blocking sets originated in the context of economic game theory in a 1956 paper by Moses Richardson. Players were identified with points in a finite projective plane and minimal winning coalitions were lines. A blocking coalition was defined as a set of points containing no line but intersecting every line. In 1958, J. R. Isbell studied these games from a non-geometric viewpoint. Jane W. DiPaola studied the minimum blocking coalitions in all the projective planes of order in 1969.

In hypergraphs

Let be a hypergraph, so that is a set of elements, and is a collection of subsets of, called edges. A blocking set of is a subset of that has nonempty intersection with each hyperedge.
Blocking sets are sometimes also called "hitting sets" or "vertex covers".
Also the term "transversal" is used, but in some contexts a transversal of is a subset of that meets each hyperedge in exactly one point.
A "two-coloring" of is a partition of
into two subsets such that no edge is monochromatic, i.e., no edge is contained entirely within or within. Now both and are blocking sets.

Complete k-arcs

In a projective plane a complete k-arc is a set of k points, no three collinear, which can not be extended to a larger arc
Theorem: Let K be a complete k-arc in Π = PG with k < q + 2. The dual in Π of the set of secant lines of K is a blocking set, B, of size k/2.

Rédei blocking sets

In any projective plane of order q, for any nontrivial blocking set B consider a line meeting B in n points. Since no line is contained in B, there must be a point, P, on this line which is not in B. The q other lines though P must each contain at least one point of B in order to be blocked. Thus, If for some line equality holds in this relation, the blocking set is called a blocking set of Rédei type and the line a Rédei line of the blocking set. Not all blocking sets are of Rédei type, but many of the smaller ones are. These sets are named after László Rédei whose monograph on Lacunary polynomials over finite fields was influential in the study of these sets.

Affine blocking sets

A set of points in the finite Desarguesian affine space that intersects every hyperplane non-trivially, i.e., every hyperplane is incident with some point of the set, is called an affine blocking set. Identify the space with by fixing a coordinate system. Then it is easily shown that the set of points lying on the coordinate axes form a blocking set of size. Jean Doyen conjectured in a 1976 Oberwolfach conference that this is the least possible size of a blocking set.
This was proved by R. E. Jamison in 1977, and independently by A. E. Brouwer, A. Schrijver in 1978 using the so-called polynomial method. Jamison proved the following general covering result from which the bound on affine blocking sets follows using duality:
Let be an dimensional vector space over. Then the number of -dimensional cosets required to cover all vectors except the zero vector is at least. Moreover, this bound is sharp.