On small blocking sets and their linearity

We prove that a small minimal blocking set of PG(2,q) is “very close” to be a linear blocking set over some subfield GF(pe)<GF(q). This implies that (i) a similar result holds in PG(n,q) for small minimal blocking sets with respect to k-dimensional subspaces (0?k?n) and (ii) most of the intervals in the interval-theorems of Sz?nyi and Sz?nyi-Weiner are empty.     

Small proper double blocking sets in Galois planes of prime order

A proper double blocking set in PG(2,p) is a set B of points such that 2?|B∩l|?(p+1)-2 for each line l. The smallest known example of a proper double blocking set in PG(2,p) for large primes p is the disjoint union of two projective triangles of side (p+3)/2; the size of this set is 3p+3. For each prime p?11 such that we construct a proper double blocking set with 3p+1 points, and for each prime p?7 we construct a proper double blocking set with 3p+2 points.     

Saturating sets in projective planes and hypergraph covers

Let ${\Pi }_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to $?\sqrt{3qlnq}?+?(\sqrt{q}+1)∕2?$. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.     

Minimal Blocking Sets in Projective Spaces of Square Order

In this paper minimal m-blocking sets of cardinality at most in projective spaces PG(n,q) of square order q, q 16, are characterized to be (t, 2(m-t-1))-cones for some t with . In particular we will find the smallest m-blocking sets that generate the whole space PG(n,q) for 2m n m.     

Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q ^k/2 of the values of d < q ^k-1. In this article we shall prove that, for q = p prime and roughly \frac38{\frac{3}{8}}-th’s of the values of d < q ^k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q ^k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over \mathbb F_p{{\mathbb F}_p} of dimension k with minimum distance d < q ^k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.     

Tight bounds for eternal dominating sets in graphs

The eternal domination number of a graph is the number of guards needed at vertices of the graph to defend the graph against any sequence of attacks at vertices. We consider the model in which at most one guard can move per attack and a guard can move across at most one edge to defend an attack. We prove that there are graphs G for which , where γ_∞(G) is the eternal domination number of G and α(G) is the independence number of G. This matches the upper bound proved by Klostermeyer and MacGillivray.     

Linear sets in finite projective spaces

In this paper linear sets of finite projective spaces are studied and the “dual” of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides “old” results, new ones are proven and some open questions are discussed.     

Conic blocking sets in Desarguesian projective planes

Define a conic blocking set to be a set of lines in a Desarguesian projective plane such that all conics meet these lines. Conic blocking sets can be used in determining if a collection of planes in projective three-space forms a flock of a quadratic cone. We discuss trivial conic blocking sets and conic blocking sets in planes of small order. We provide a construction for conic blocking sets in planes of non-prime order, and we make additional comments about the structure of these conic blocking sets in certain planes of even order.     

Linear programming bounds for codes in infinite projective spaces

We develop a technique for improving the universal linear programming bounds on the cardinality and the minimum distance of codes in projective spaces . We firstly investigate test functions P_j(m,n,s) having the property that P_j(m,n,s)<0 for somej if and only if the corresponding universal linear programming bound can be further improved by linear programming. Then we describe a method for improving the universal bounds. We also investigate the possibilities for attaining the first universal bounds.     

Weakly open sets in the unit ball of the projective tensor product of Banach spaces

A Banach space is said to have the diameter two property if every non-empty relatively weakly open subset of its unit ball has diameter two. We prove that the projective tensor product of two Banach spaces whose centralizer is infinite-dimensional has the diameter two property. The same statement also holds for if the centralizer of X is infinite-dimensional and the unit sphere of Y^? contains an element of numerical index one. We provide examples of classical Banach spaces satisfying the assumptions of the results. If K is any infinite compact Hausdorff topological space, then has the diameter two property for any nonzero Banach space Y. We also provide a result on the diameter two property for the injective tensor product.     

On the number of pairwise non-isomorphic minimal blocking sets in PG(2, q)

In this paper we examine whether the number of pairwise non-isomorphic minimal blocking sets in PG(2, q) of a certain size is larger than polynomial. Our main result is that there are more than polynomial pairwise non-isomorphic minimal blocking sets for any size in the intervals [2q−1, 3q−4] for q odd and for q square. We can also prove a similar result for certain values of the intervals and .      

Blocking sets in line Grassmannians

In this paper the most natural questions concerning the blocking sets in the line Grassmannian of PG(n,q) are partially answered. In particular, the following Bose-Burton type theorems are proved: if n is odd or n=4, then the blocking sets of minimum size are precisely the linear complexes with singular subspace of minimum dimension.     

Lower bounds on the cardinality of the maximum matchings of planar graphs

Lower bounds on the cardinality of the maximum matchings of planar graphs, with a constraint on the minimum degree, are established in terms of a linear polynomial of the number of vertices. The bounds depend upon the minimum degree and the connectivity of graphs. Some examples are given which show that all the lower bounds are best possible in the sense that neither the coefficients nor the constant terms can be improved.     

On the intersection problem for linear sets in the projective line

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.     

Proper minimal points of nonconvex sets in Banach spaces in terms of the limiting normal cone

In this paper, proper minimal elements of a given nonconvex set in a real ordered Banach space are defined utilizing the limiting (Mordukhovich) normal cone. The newly defined points are called limiting proper minimal (LPM) points. It is proved that each LPM is a proper minimal in the sense of Borwein under some assumptions. The converse holds in Asplund spaces. The relation of LPM points with Benson, Henig, super and proximal proper minimal points are established. Under appropriate assumptions, it is proved that the set of robust elements is a subset of the set of LPM points, and the set of LPM points is dense in that of minimal points. Another part of the paper is devoted to scalarization-based and distance function-based characterizations of the LPM points. The paper is closed by some results about LPM solutions of a set-valued optimization problem via variational analysis tools. Clarifying examples are given in addition to the theoretical results.     

On the cardinality of blocking sets in PG(2,q)

Denote by PG(2,q) the finite desarguesian projective plane of order q, where q=p^h, p a prime, q>2. We define the function m(q) as follows: m(q)=q, if q is a square; m(q)=(q+1)/2, if q is a prime; m(q)=p^h–d, if q=p^h with h an odd integer, where d denotes the greatest divisor of h different from h. The following theorem is proved: For any integer k with q+m(q)+1 k q²–m(q), there exists a blocking set in PG(2,q) having exactly k elements.To Professor Adriano Barlotti on his 60th birthday.Research partially supported by G.N.S.A.G.A. (CNR)     

On πgs-closed sets in topological spaces

Summary A new class of sets called πgs-closed sets is introduced and its properties are studied. Moreover the notions of πgs-T_1/2 spaces and πgs-continuity are introduced.