János Barát

Directed Path-width and Monotonicity in Digraph Searching

Directed path-width was defined by Reed, Thomas and Seymour around 1995. The author and P. Hajnal defined a cops-and-robber game on digraphs in 2000. We prove that the two notions are closely related and for any digraph D, the corresponding graph parameters differ by at most one. The result is achieved using the mixed-search technique developed by Bienstock and Seymour. A search is called monotone, in which the robbers territory never increases. We show that there is a mixed-search of D with k cops if and only if there is a monotone mixed-search with k cops. For our cops-and-robber game we get a slightly weaker result: the monotonicity can be guaranteed by using at most one extra cop.

Minimal Blocking Sets in PG( 2 , 8 ) and Maximal Partial Spreads in PG( 3 , 8 )

We prove that PG(2, 8) does not contain minimal blocking sets of size 14. Using this result we prove that 58 is the largest size for a maximal partial spread of PG(3, 8). This supports the conjecture that q2−q+ 2 is the largest size for a maximal partial spread of PG(3, q), q>7.

Elementary proof techniques for the maximum number of islands

Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. It is a fundamental property that two islands are either containing or disjoint. Czedli determined the maximum number of rectangular islands. Pluhar solved the same problem for bricks, and Horvath, Nemeth and Pluhar for triangular islands. Here, we give a much shorter proof for these results, and also for new, analogous results on toroidal and some other boards.

What is on his mind

Abstract We survey 19 questions and conjectures of Carsten Thomassen, most of which remain open.

Empty pentagons in point sets with collinearities

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.

Multiple Blocking Sets in PG( n , q ), n > 3

This article discusses minimal s-fold blocking sets B in PG (n, q), q = ph, p prime, q > 661, n > 3, of size |B| > sq + cpq2/3 - (s - 1) (s - 2)/2 (s > min (cpq1/6, q1/4/2)). It is shown that these s-fold blocking sets contain the disjoint union of a collection of s lines and/or Baer subplanes. To obtain these results, we extend results of Blokhuis–Storme–Szönyi on s-fold blocking sets in PG(2, q) to s-fold blocking sets having points to which a multiplicity is given. Then the results in PG(n, q), n ≥ 3, are obtained using projection arguments. The results of this article also improve results of Hamada and Helleseth on codes meeting the Griesmer bound.

On complete caps in the projective geometries over\(\mathbb{F}_3 \)

It is known that the largest size of cap in PG(5, 3) is 56, but very little is known about complete caps of smaller size; the previously known complete caps withk < 56 all had size at most 43. In this paper we construct complete 48-caps and show that any 53-cap is extendable to a 56-cap. From this last result, we derive new upper bounds on the largest size of cap in PG(r, 3) forr ≥ 6. The results are obtained from a blend of geometric and coding theoretic techniques.

Operations Which Preserve Path-Width at Most Two

The number of excluded minors for the class of graphs with path-width at most two is very large. To give a practical characterization of the obstructions, we introduce some operations which preserve path-width at most two. We give a list of ten graphs such that any graph with path-width more than two can be reduced – by taking minors and applying our operations – to one of the graphs on our list. We think that our operations and excluded substructures give a far more transparent description of the class of graphs with path-width at most two than Kinnersley and Langstons characterization by 110 excluded minors (see [4]).

Large Bd-free and union-free subfamilies

Abstract For a property Γ and a family of sets F , let f ( F , Γ ) be the size of the largest subfamily of F having property Γ. For a positive integer m, let f ( m , Γ ) be the minimum of f ( F , Γ ) over all families of size m. A family F is said to be B d -free if it has no subfamily F ′ = { F I : I ⊆ [ d ] } of 2 d distinct sets such that for every I , J ⊆ [ d ] , both F I ∪ F J = F I ∪ J and F I ∩ F J = F I ∩ J hold. A family F is a-union free if F 1 ∪ ⋯ ∪ F a ≠ F a + 1 whenever F 1 , … , F a + 1 are distinct sets in F . We verify a conjecture of Erdős and Shelah that f ( m , B 2 -free ) = Θ ( m 2 / 3 ) . We also obtain lower and upper bounds for f ( m , B d -free ) and f ( m , a -union free ) .