Tamás Héger

• Doctor of Philosophy
• Budapest University of Technology and Economics

Let be a prime power and be a natural number. What are the possible cardinalities of point sets in a projective plane of order , which do not intersect any line at exactly points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show...

Let $q$ be a prime power and $k$ be a natural number. What are the possible cardinalities of point sets ${S}$ in a projective plane of order $q$, which do not intersect any line at exactly $k$ points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In...

We improve the known bounds on the size of identifying codes in the collinearity graphs of generalized quadrangles.

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few...

In this article, we summarize our research results on the topic of spontaneous emergence of intelligence. Many agents are sent to an artificial world, which is arbitrarily parametrizable. The agents initially know nothing about the world. Their only ability is the remembrance, that is, the use of experience which comes from the events that happened...

We provide new values of the bipartite Ramsey number RB(C4,K1,n) using induced subgraphs of the incidence graph of a projective plane. The approach, based on deleting subplanes of projective planes, has been used in related extremal problems and allows us to unify previous results and extend them. More importantly, using deep stability results on 2...

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$-element Galois field which is linear in both $q$ and $k$, hen...

We investigate the upper chromatic number of the hypergraph formed by the points and the k‐dimensional subspaces of PG(n,q); that is, the most number of colors that can be used to color the points so that every k‐subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color,...

We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with t...

The main purpose of this paper is to find double blocking sets in PG(2,q) of size less than 3q, in particular when q is prime. To this end, we study double blocking sets in PG(2,q) of size 3q−1 admitting at least two (q−1)-secants. We derive some structural properties of these and show that they cannot have three (q−1)-secants. This yields that one...

The main purpose of this paper is to find double blocking sets in $\mathrm{PG}(2,q)$ of size less than $3q$, in particular when $q$ is prime. To this end, we study double blocking sets in $\mathrm{PG}(2,q)$ of size $3q-1$ admitting at least two $(q-1)$-secants. We derive some structural properties of these and show that they cannot have three $(q-1...

In this paper the metric dimension of (the incidence graphs of) particular partial linear spaces is considered. We prove that the metric dimension of an affine plane of order $q\geq13$ is $3q-4$ and describe all resolving sets of that size if $q\geq 23$. The metric dimension of a biaffine plane (also called a flag-type elliptic semiplane) of order...

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order q(2) of size q(2) + 2q + 2 admitting 1-,2-,3-,4-, (q + 1)- and (q + 2)-secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order q(2) of size at most 4q(...

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order $q>81$ is smaller than $2q+2[\sqrt{q}]+2$ (i.e., twice the size o...

We describe small dominating sets of the incidence graphs of finite projec- tive planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dom- inating set in a projective plane of order q > 81 is smaller than 2q +2q^(1/2)+2 (i.e., twice the size o...

A $t$-semiarc is a pointset ${\cal S}_t$ with the property that the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. Some characterization theorems for small semiarcs with large collinear subsets in $\mathrm{PG}(2,q)$ are given.

We consider the following $q$-analog of the basic combinatorial search problem: let $q$ be a prime power and $\GF(q)$ the finite field of $q$ elements. Let $V$ denote an $n$-dimensional vector space over $\GF(q)$ and let $\mathbf{v}$ be an unknown 1-dimensional subspace of $V$. We will be interested in determining the minimum number of queries that...

AbstractA twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd,...

In 1967, Brown constructed small k-regular graphs of girth six as induced subgraphs of the incidence graph of a projective plane of order q, q >= k. Examining the construction method, we prove that starting from PG(2. q), q = p(h), p prime, there are no other constructions using this idea resulting in a (q + 1 - t)-regular graph of girth six than t...

In the paper we consider some constructions of (k,6)-graphs that are isomorphic to an induced subgraph of the incidence graph of a finite projective plane, and present some unifying concepts. Also, we obtain new bounds on and exact values of Zarankiewicz numbers, mainly when the parameters are close to those of a design.

We show that the metric dimension of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in $\PG(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of $\PG(2,q)$, $...

Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified), for any elements a1,…,an of GF(q), there are distinct field elements b1,…,bn such that a1b1+⋯+anbn=0. This implies the classification of hyperplanes lying in the union of the hyperplanes Xi=Xj in a vector space over GF(q),...

Small k-regular graphs of girth g where g=6,8,12 are obtained as subgraphs of minimal cages. More precisely, we obtain (k,6)-graphs on 2(kq−1) vertices, (k,8)-graphs on 2k(q2−1) vertices and (k,12)-graphs on 2kq2(q2−1), where q is a prime power and k is a positive integer such that q≥k≥3. Some of these graphs have the smallest number of vertices kn...

We give new constructions for k-regular graphs of girth 6, 8 and 12 with a small number of vertices. The key idea is to start with a generalized n-gon and delete some lines and points to decrease the valency of the incidence graph.