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" (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  2"n". 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 3"m" - 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 (3"q" + 2)/2. "Theorem": In PG(2,"p"), with "p  a prime, there exists a projective triad of side (p" + 1)/2 which is a blocking set of size (3"p"+ 1)/2. Size. One typically searches for small blocking sets. The minimum size of a blocking set of formula_1 is called formula_2. In the Desarguesian projective plane of order "q", PG(2,"q"), 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 formula_4 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 formula_5 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(2,"p"), "p" a prime, has size at least 3("p" + 1)/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 formula_6 in 1969. In hypergraphs. Let formula_7 be a hypergraph, so that formula_8 is a set of elements, and formula_9 is a collection of subsets of formula_8, called (hyper)edges. A blocking set of formula_1 is a subset formula_12 of formula_8 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 formula_1 is  a subset formula_15 of formula_8 that meets each hyperedge in exactly one point. A "two-coloring" of formula_1 is a partition formula_18 of formula_8 into two subsets (color classes) such that no edge is monochromatic, i.e., no edge is contained entirely within formula_20 or within formula_21. Now both formula_20 and formula_21 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 (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. Rédei blocking sets. 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, formula_24 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"). 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 formula_25 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 formula_26 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 formula_27. 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 formula_28 be an formula_29 dimensional vector space over formula_30. Then the number of formula_31-dimensional cosets required to cover all vectors except the zero vector is at least formula_32. Moreover, this bound is sharp."