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.

It has been conjectured by D. R. Stinson that an incomplete Room square \((n, s)\)-IRS exists if and only if \(n\) and \(s\) are both odd and \(n \geq 3s + 2\), except for the nonexistent case \((n, s) = (5, 1)\). In this paper we shall improve the known results and show that the conjecture is true except for \(45\) pairs \((n, s)\) for which the existence of an \((n, s)\)-IRS remains undecided.