This note is part of a general programme to classify the automorphism groups of nite linear spaces. There have been a number of contributions to this programme including two recent surveys, 8, 3]. One of the most signiicant contributions was the classiication of ag-transitive linear spaces, 1]. Since then the eeort has been to classify the line-transitive examples. These fall naturally into two...

Abstract: A Latin square arrangement is an arrangement of s symbols in s rows and s columns, such that every symbol occurs once in each row and each column. When two Latin squares of same order superimposed on one another, then in the resultant array every ordered pair of symbols occurs exactly once, then the two Latin squares are said to be orthogonal. A frequency square M of type F (n; λ) is ...

Let (W, C) be an m-cycle system of order n and let Ω ⊂ W , |Ω| = v < n. We say that a path design (Ω,P) of order v and block size s (2 ≤ s ≤ m− 1) is embedded in (W, C) if for every p ∈ P there is an m-cycle c = (a1, a2, . . . , am) ∈ C such that: (1) p = [ak, ak+1, . . . , ak+s−1] for some k ∈ {1, 2, . . . ,m} (i.e. the (s− 1)-path p occurs in the m-cycle c); and (2) ak−1, ak+s ∈ Ω. Note that ...

A theorem of Venkov says that each nontrivial shell of an extremal even unimodular lattice in R with 24 | n is a spherical 11design. It is a difficult open question whether there exists any 12-design among them. In the first part of this paper, we consider the following problem: When do all shells of an even unimodular lattice become 12designs? We show that this does not happen in many cases, t...

There are four resolvable Steiner triple systems on fifteen elements. Some generalizations of these systems are presented here. 0. Introduction. The following definition is standard (see [2]). Definition. Let V be a finite set, v = #(V ) be the number of elements of V , let (V/3) (resp. (V/2)) be the set of all unordered triads (resp. pairs) of distinct elements of V . A Steiner triple system o...

With any G-symmetric graph admitting a nontrivial G-invariant partition B, we may associate a natural ‘‘cross-sectional’’ geometry, namely the 1-design DðBÞ 1⁄4 ðB; BðBÞ; IÞ in which IC for 2B and C 2 BðBÞ if and only if is adjacent to at least one vertex in C, where B2B and BðBÞ is the neighbourhood of B in the quotient graph B of with respect to B. In a vast number of cases, the dual 1-design...

For a block design D, a series of block intersection graphs Gi, or i-BIG(D), i = 0, . . . , k is defined in which the vertices are the blocks of D, with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α-set. Let G be an r-regular graph and c be a proper (r + 1)-co...

A packing of Kn with copies of C4 (the cycle of length 4), is an ordered triple (V, C, L), where V is the vertex set of the complete graph Kn, C is a collection of edge-disjoint copies of C4, and L is the set of edges not belonging to a block of C. The number n is called the order of the packing and the set of unused edges L is called the leave. If C is as large as possible, then (V, C, L) is c...

A packing of Kn with copies of C4 (the cycle of length 4), is an ordered triple (V, C, L), where V is the vertex set of the complete graph Kn, C is a collection of edge-disjoint copies of C4, and L is the set of edges not belonging to a block of C. The number n is called the order of the packing and the set of unused edges L is called the leave. If C is as large as possible, then (V, C, L) is c...

Given a family F of r-graphs, let ex(n,F) be the maximum number of edges in an n vertex r-graph containing no member of F . Let C 4 denote the family of r-graphs with distinct edges A,B, C, D, such that A ∩ B = C ∩ D = ∅, A ∪ B = C ∪ D. For s1 ≤ · · · ≤ sr, let K(s1, . . . , sr) be the complete r-partite r-graph with parts of sizes s1, . . . , sr. Füredi conjectured over 15 years ago that ex(n,...