A star \(S_q\), with \(q\) edges, is a complete bipartite graph \(K_{1,q}\). Two figures of the complete graph \(K_n\) on a given set of \(k\) vertices are compatible if they are edge-disjoint, and a configuration is a set of pairwise compatible figures. In this paper, we take stars as our figures. A configuration \(C\) is said to be maximal if there is no figure (star) \(f \notin C\) such that \(\{f\} \cup C\) is also a configuration. The size of a configuration \(F\), denoted by \(|F|\), is the number of its figures. Let \(\text{Spec}(n, q)\) (or simply \(\text{Spec}(n)\)) denote the set of all sizes such that there exists a maximal configuration of stars with this size. In this paper, we completely determine \(\text{Spec}(n)\), the spectrum of maximal configurations of stars. As a special case, when \(n\) is an order of a star system, we obtain the spectrum of maximal partial star systems.

It is proved in this paper that for \(\lambda = 4\) and \(5\), the necessary conditions for the existence of a simple \(B(4, \lambda; v)\) are also sufficient. It is also proved that for \(\lambda = 4\) and \(5\), the necessary conditions for the existence of an indecomposable simple \(B(4, \lambda; v)\) are also sufficient, with the unique exception \((v, \lambda) = (7, 4)\) and \(10\) possible exceptions.

Let \(S\) and \(T\) be sets with \(|S| = m\) and \(|T| = n\). Let \(S_3, S_2\) and \(T_3, T_2\) be the sets of all \(3\)-subsets (\(2\)-subsets) of \(S\) and \(T\), respectively. Define \(Q((m, 2, 3), (n, 2, 3))\) as the smallest subset of \(S_2 \times T_2\) needed to cover all elements of \(S_3 \times T_3\). A more general version of this problem is initially defined, but the bulk of the investigation is devoted to studying this number. Its property as a lower bound for a planar crossing number is the reason for this focus.

Under some assumptions on the incidence matrices of symmetric designs, we prove a non-existence theorem for symmetric designs. The approach generalizes Wilbrink’s result on difference sets \([7]\).

In this paper, we derive some inequalities which the parameters of a two-symbol balanced array \(T\) (\(B\)-array) of strength four must satisfy for \(T\) to exist.

This paper considers Latin squares of order \(n\) having \(0, 1, \ldots, n-1\) down the main diagonal and in which the back diagonal is a permutation of these symbols (diagonal squares). It is an open question whether or not such a square which is self-orthogonal (i.e., orthogonal to its transpose) exists for order \(10\). We consider two possible constraints on the general concept: self-conjugate squares and strongly symmetric squares. We show that relative to each of these constraints, a corresponding self-orthogonal diagonal Latin square of order \(10\) does not exist. However, it is easy to construct self-orthogonal diagonal Latin squares of orders \(8\) and \(12\) which satisfy each of the constraints respectively.