Let \(x_1, x_2, \ldots, x_v\) be commuting indeterminates over the integers. We say an \(v \times v \times v \ldots \times v \) n-dimensional matrix is a proper \(v\)-dimensional orthogonal design of order \(v\) and type \((s_1, s_2, \ldots, s_r)\) (written \(\mathrm{OD}^n(s_1, s_2, \ldots, s_r)\)) on the indeterminates \(x_1, x_2, \ldots, x_r\) if every 2-dimensional axis-normal submatrix is an \(\mathrm{OD} (s_1, s_2, \ldots, s_r)\) of order \(v\) on the indeterminates \(x_1, x_2, \ldots, x_r\). Constructions for proper \(\mathrm{OD}^n(1^2)\) of order 2 and \(\mathrm{OD}^n(1^4)\) of order 4 are given in J. Seberry (1980) and J. Hammer and J. Seberry (1979, 1981a), respectively. This paper contains simple constructions for proper \(\mathrm{OD}^n(1^{2})\), \(\mathrm{OD}^n(1^{4})\), and \(\mathrm{OD}^n(1^{ 8})\) of orders 2, 4, and 8, respectively. Prior to this paper no proper higher dimensional OD on more than 4 indeterminates was known.

Bondy and Fan recently conjectured that if we associate non-negative real weights to the edges of a graph so that the sum of the edge weights is \(W\), then the graph contains a path whose weight is at least \(\frac{2W}{n}\). We prove this conjecture.

Let \(H(V, E)\) be an \(r\)-uniform hypergraph. Let \(A \subset V\) be a subset of vertices and define \(\deg_H(A) = |\{e \in E : A \subset e\}|\).

We say that \(H\) is \((k, m)\)-divisible if for every \(k\)-subset \(A\) of  \(V(H)\), \(\deg_H(A) \equiv 0 \pmod{m}\). (We assume that \(1 \leq k < r\)).

Given positive integers \(r \geq 2\), \(k \geq 1\) and \(q\) a prime power, we prove that if \(H\) is an \(r\)-uniform hypergraph and \(|E| > (q-1) \binom{\mid V \mid}{k} \), then \(H\) contains a nontrivial subhypergraph \(F\) which is \((k, q)\)-divisible.

It is well known that there exist complete \(k\)-caps in \(\mathrm{PG}(3,q)\) with \(k \geq \frac{q^2+q+4}{2}\) and it is still unknown whether or not complete \(k\)-caps of size \(k < \frac{q^2+q+4}{2}\) and \(q\) odd exist. In this paper sufficient conditions for the existence of complete \(k\)-caps in \(\mathrm{PG}(3,q)\), for good \(q \geq 7\) and \(k < \frac{q^2+q+4}{2}\), are established and a class of such complete caps is constructed.