Definition[edit]

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.^[1]

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.

History[edit]

Blocking sets originated^[8] in the context of economic game theory in a 1956 paper by Moses Richardson.^[9] 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^[10] studied these games from a non-geometric viewpoint. Jane W. DiPaola studied the minimum blocking coalitions in all the projective planes of order $\leq 9$ in 1969.^[11]

In hypergraphs[edit]

Let $H=(X,E)$ be a hypergraph, so that $X$ is a set of elements, and $E$ is a collection of subsets of $X$ , called (hyper)edges. A blocking set of $H$ is a subset $S$ of $X$ 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 $H$ is a subset $T$ of $X$ that meets each hyperedge in exactly one point.

A "two-coloring" of $H$ is a partition $\{C,D\}$ of $X$ into two subsets (color classes) such that no edge is monochromatic, i.e., no edge is contained entirely within $C$ or within $D$ . Now both $C$ and $D$ are blocking sets.

Complete k-arcs[edit]

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 (thus, every point not on the arc is on a secant line of the arc–a line meeting the arc in two points.)

Theorem: Let K be a complete k-arc in Π = PG(2,q) with k < q + 2. The dual in Π of the set of secant lines of K is a blocking set, B, of size k(k - 1)/2.^[12]

Rédei blocking sets[edit]

In any projective plane of order q, for any nontrivial blocking set B (with b = |B|, the size of the blocking set) 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, $b\geq n+q.$ 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 (note that n will be the largest number of collinear points in B).^[13] 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.^[14]

Affine blocking sets[edit]

A set of points in the finite Desarguesian affine space $AG(n,q)$ 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 $\mathbb {F} _{q}^{n}$ 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 $1+n(q-1)$ . 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,^[15] and independently by A. E. Brouwer, A. Schrijver in 1978 ^[16] 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 $V$ be an $n$ dimensional vector space over $\mathbb {F} _{q}$ . Then the number of $k$ -dimensional cosets required to cover all vectors except the zero vector is at least $q^{n-k}-1+k(q-1)$ . Moreover, this bound is sharp.

Barwick, Susan; Ebert, Gary (2008), Unitals in Projective Planes, New York: Springer, :10.1007/978-0-387-76366-8, ISBN 978-0-387-76364-4, ISSN 1439-7382

doi

Graphs and hypergraphs, North-Holland, Amsterdam, 1973. (Defines $\tau (H)$ .)

C. Berge

P. Duchet, Hypergraphs, Chapter 7 in: Handbook of Combinatorics, North-Holland, Amsterdam, 1995.

(1979), Projective Geometries over Finite Fields, Oxford: Oxford University Press, ISBN 978-0-19-853526-3

Hirschfeld, J.W.P.

Holder, Leanne D. (2001), Blocking Sets of Conics, Ph.D. thesis, University of Colorado Denver

De Beule, Jan; Storme, Leo (2011), , Nova Science Publishers, ISBN 978-1-61209-523-3, archived from the original on 2016-01-29, retrieved 2016-01-23