On the intersection problem for linear sets in the projective line

Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.

Inequivalence of skew Hadamard difference sets and triple intersection numbers modulo a prime

Recently, Feng and Xiang \cite{FX113} found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show that their construction produces infinitely many skew Hadamard difference sets inequivalent to the Paley difference sets.

Intersections of shifted sets

We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the following: If $A=\{a_n\}$ is such that $a_n=o(n^{k/k-1})$, then the $k$-recurrence set $R_k(A)=\{x\mid |A\cap(A+x)|\ge k\}$ contains the distance sets of arbitrarily large finite sets.

The number of k-intersections of an intersecting family of r-sets

The Erdos-Ko-Rado theorem tells us how large an intersecting family of r-sets from an n-set can be, while results due to Lovasz and Tuza give bounds on the number of singletons that can occur as pairwise intersections of sets from such a family. We consider a natural generalization of these problems. Given an intersecting family of r-sets from an n-set and 1\leq k \leq r, how many k-sets can occur as pairwise intersections of sets from the family? For k=r and k=1 this reduces to the problems described above. We answer this question exactly for all values of k and r, when n is sufficiently larg…

Conference matrices with maximum excess and two-intersection sets

A two-intersection set with parameters $(j;α,β)$ for a block design is a $j$-subset of the point set of the design, which intersects every block in $α$ or $β$ points. In this paper, we show the existence of a two-intersection set with parameters $(2m^2-m+1;m^2-m,m^2)$ for the block design obtained from translations of the set of nonzero squares in the finite field of order $q=4m^2+1$. As an application, we give a construction of conference matrices with maximum excess based on the two-intersection sets.

Synthetic projective lines, geometric closure and AB-sets

In this note, we introduce a new approach to abstract ``synthetic'' projective lines. We discuss various aspects of our approach, and compare these aspects with the classical one. A number of intriguing questions arise. Amongst these aspects, we discuss geometric closures, and introduce automorphism-blocking sets. We also construct a number of (counter) examples in infinite cases.

On MPS construction of blocking sets in projective spaces: a generalization

In this paper we provide a generalization of the MPS construction of blocking sets of $PG(r,q^n)$ using subspaces of dimension $s\leq n-2$. By this construction, we determine a new non-planar example in $PG(3,q^6)$.

具有二重和三重交点的极小加一生成线排列

Minimal plus-one generated line arrangements with double and triple intersection points

该论文在弱组合学框架下，完整分类了复射影平面中具有二重和三重交点的极小加一生成线排列。

Almost all string graphs are intersection graphs of plane convex sets

A {\em string graph} is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of {\em almost all} string graphs on $n$ vertices can be partitioned into {\em five} cliques such that some pair of them is not connected by any edge ($n\rightarrow\infty$). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollar…

Ring structure, uniform expressions and intersection homology

Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any resolution $X_r\to X$. This paper uses uniform (and non-uniform) expressions to define what is believed to be the usual intersection homology (and its local-global variant) of a convex polytope (or a projective toric variety). Such expressions are generated by the facets, and so may lead to necessary numerical conditions on the flag vector. Most of the conc…

Sets of mutually orthogoval projective and affine planes

A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review known results that are equivalent to sets of more than two mutually orthogoval planes. We also discuss the connection between sets of mutually orthogoval planes and covering arrays.

Coloring intersection graphs of arc-connected sets in the plane

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.

Complete intersection vanishing ideals on sets of clutter type over finite fields

In this paper we give a classification of complete intersection vanishing ideals on parameterized sets of clutter type over finite fields.

Triangle-free geometric intersection graphs with no large independent sets

It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.