Related papers: Small transitive homogeneous $3$-$(v,\{4,6\},1)$ d…

The full classification of Riemannian $3$-symmetric spaces is presented. Up to Riemannian products the main building blocks consist in (possibly symmetric) spaces with semisimple isometry group, nilpotent Lie groups of step at most $2$ and…

In this paper we complete a classification of finite linear spaces $\cS$ with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders $4,7,…

Let $G$ be an additive group of order $v$. A $k$-element subset $D$ of $G$ is called a $(v, k, \lambda, t)$-almost difference set if the expressions $gh^{-1}$, for $g$ and $h$ in $D$, represent $t$ of the non-identity elements in $G$…

Given an STS(v), we ask if there is a permutation of the points of the design such that no $\ell$ consecutive points in this permutation contain a block of the design. Results are obtained in the cases $\ell = 3,4$.

Let 1_k 0_l denote the (k+l)\times 1 column of k 1's above l 0's. Let q. (1_k 0_l) $ denote the (k+l)xq matrix with q copies of the column 1_k0_l. A 2-design S_{\lambda}(2,3,v) can be defined as a vx(\lambda/3)\binom{v}{2} (0,1)-matrix with…

A two-dimensional simplicial complex is called $d$-{\em regular} if every edge of it is contained in exactly $d$ distinct triangles. It is called $\epsilon$-expanding if its up-down two-dimensional random walk has a normalized maximal…

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…

A combinatorial block design $D$ is called $3$-pyramidal if there exists a subgroup $G$ of $\mbox{Aut}(D)$ fixing $3$ points and acting regularly on the other points. If this happens, we say that the design is $3$-pyramidal under $G$. In…

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block…

A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of…

A $t\text{-}(n,K,\lambda;q)$ design, also called the $q$-analog of a $t$-wise balanced design, is a set ${\mathcal B}$ of subspaces with dimensions contained in $K$ of the $n$-dimensional vector space ${\mathbb F}_q^n$ over the finite field…

Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…

A graph $\Ga$ is $G$-symmetric if $\Ga$ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Ga$, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is…

A {\em cross-free} set of size $m$ in a Steiner triple system $(V,{\cal{B}})$ is three pairwise disjoint $m$-element subsets $X_1,X_2,X_3\subset V$ such that no $B\in {\cal{B}}$ intersects all the three $X_i$-s. We conjecture that for every…

The automorphism group $\Sigma$ of a compact topological projective plane with a $16$-dimensional point space is a locally compact group. If the dimension of $\Sigma$ is at least $29$, then $\Sigma$ is known to be a Lie group. For the…

A $t$-$(n,k,\lambda)$ design over $\F_q$ is a collection of $k$-dimensional subspaces of $\F_q^n$, called blocks, such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly $\lambda$ blocks. Such $t$-designs over $\F_q$ are…

An Orthogonally resolvable Matching Design OMD$(n, k)$ is a partition of the edges the complete graph $K_n$ into matchings of size $k$, called blocks, such that the blocks can be resolved in two different ways. Such a design can be…

A finite linear space is a finite set of points and lines, where any two points lie on a unique line. Well known examples include projective planes. This project focuses on linear spaces which admit certain types of symmetries. Symmetries…

A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the complete directed graph $\vec{K_v}$ into transitive triples. A $v$-good sequencing of a DTS$(v)$ is a permutation of the points of the design, say $[x_1 \; \cdots…

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear…