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.     

Subdifferential properties for a class of minimal time functions with moving target sets in normed spaces

In a normed vector space, we study the minimal time function determined by a moving target set and a differential inclusion, where the set-valued mapping involved has constant values of a bounded closed convex set U. After establishing a characterization of ?-subdifferential of the minimal time function, we obtain that the limiting subdifferential of the minimal time function is representable by virtue of the corresponding normal cones of sublevel sets of the function and level or sublevel sets of the support function of U. The known results require the set U to have the origin as an interior point and the target set is a fixed set.     

Large blocking sets in PG(2,q2)

Minimal blocking sets in $\mathrm{PG}(2,q^2)$ have size at most $q^3+1$. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most $q^3+1-(p-3)/2$, if $q=p$, $p\ge 67$, or $q=p^h$, $p>7$, $h>1$. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets).     

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.     

Largest minimal blocking sets in PG(2,8)

Bruen and Thas proved that the size of a large minimal blocking set is bounded by . Hence, if q = 8, then the maximal possible size is 23. Since 8 is not a square, it was conjectured that a minimal blocking 23‐set does not exist in PG(2,8). We show that this is not the case, and construct such a set. We prove that this is combinatorially unique. We also complete the spectrum problem of minimal blocking sets for PG(2,8) by showing a minimal blocking 22‐set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 162–169, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10035     

Exact separation theorem for closed sets in Asplund spaces

In this paper, in terms of the Fréchet normal cone, we establish exact separation results for finitely many disjoint closed sets in an Asplund space, which supplement the extremal principle and some fuzzy separation theorems. As an application, we provide a new optimality condition for a constraint optimization problem in terms of Fréchet subdifferential and Fréchet normal cone.     

On a family of minimal blocking sets of the Hermitian surface

An infinite family of minimal blocking sets of ??(3, q²) is constructed for even q, with links to Ceva configurations. Copyright © 2011 John Wiley & Sons, Ltd 19:313‐316, 2011.     

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.     

Resolving sets for higher dimensional projective spaces

Lower and upper bounds on the size of resolving sets for the point-hyperplane incidence graph of the finite projective space $\mathrm{PG}(n,q)$ are presented.     

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.     

Topological structure of solution sets in Fréchet spaces: The projective limit approach

In this paper we show that the solution set of certain Volterra inclusions defined between Fréchet spaces is a continuum. The proof relies on results in Banach spaces and on viewing a Fréchet space as a projective limit of a sequence of Banach spaces.