A dependence system on a set \(S\) is defined by an operator \(f\), a function on the power set of \(S\) which is extensive (\(A\) is included in \(f(A)\)) and monotone (if \(A\) is included in \(B\), then \(f(A)\) is included in \(f(B)\)). In this paper, the structure of the set \(F(S)\) of all dependence systems on a given set \(S\) is studied. The partially ordered set of operators (\(f < g\) if for every set \(A\), \(f(A)\) is included in \(g(A)\)) is a bounded, complete, completely distributive, atomic, and dual atomic lattice with an involution. It is shown that every operator is a join of transitive operators (usually called closure operators, operators which are idempotent \(ff = f\)). The study of the class of transitive operators that join-generate all operators makes it possible to express \(F(n)\) (the cardinality of the set \(F(S)\) of all operators on a set \(S\) with \(n\) elements) by the Dedekind number \(D(n)\). The formula has interesting consequences for dependence system theory.

Let \(p(k)\) (\(q(k)\)) be the smallest number such that in the projective plane, every simple arrangement of \(n \geq p(k)\) (\( \geq q(k)\)) straight lines (pseudolines) contains \(k\) lines which determine a \(k\)-gonal region. The values \(p(6) = q(6) = 9\) are determined and the existence of \(q(k) (\geq p(k))\) is proved.

We introduce a complex version of Golay sequences and show how these may be applied to obtain new Hadamard matrices and complex Hadamard matrices. We also show how to modify the Goethals-Seidel array so that it can be used with complex sequences.

In this paper, we improve the best known algorithm on symmetric equivalence of Hadamard matrices by considering the eigenvalues of symmetric Hadamard matrices. As a byproduct, the eigenvalues of skew Hadamard matrices are also discussed.

The spectrum for \(k\)-perfect \(3k\)-cycle systems is considered here for arbitrary \(k \not\equiv 0 \pmod{3}\). Previously, the spectrum when \(k = 2\) was dealt with by Lindner, Phelps, and Rodger, and that for \(k = 3\) by the current authors. Here, when \(k \equiv 1\) or \(5 \pmod{6}\) and \(6k + 1\) is prime, we show that the spectrum for \(k\)-perfect \(3k\)-cycle systems includes all positive integers congruent to \(1 \pmod{6k}\) (except possibly the isolated case \(12k + 1\)). We also complete the spectrum for \(k = 4\) and \(5\) (except possibly for one isolated case when \(k = 5\)), and deal with other specific small values of \(k\).

An efficient dominating set \(S\) for a graph \(G\) is a set of vertices such that every vertex in \(G\) is in the closed neighborhood of exactly one vertex in \(S\). If \(G\) is connected and has no vertices of degree one, then \(G\) has a spanning tree which has an efficient dominating set. An \(O(n)\) algorithm for finding such a spanning tree and its efficient dominating set is given.

Numbers similar to the van der Waerden numbers \(w(n)\) are studied, where the class of arithmetic progressions is replaced by certain larger classes. If \(\mathcal{A}’\) is such a larger class, we define \(w'(n)\) to be the least positive integer such that every \(2\)-coloring of \(\{1, 2, \ldots, w'(n)\}\) will contain a monochromatic member of \(\mathcal{A}’\). We consider sequences of positive integers \(\{x_1, \ldots, x_n\}\) which satisfy \(x_i = a_i x_{i-1} + b_i x_{i-2}\) for \(i = 3, \ldots, n\) with various restrictions placed on the \(a_i\) and \(b_i\). Upper bounds are given for the corresponding functions \(w'(n)\). Further, it is shown that the existence of somewhat stronger bounds on \(w'(n)\) would imply certain bounds for \(w(n)\).

For any graph \(G\), and all \(s = 2^k\), we show there is a partition of the vertex set of \(G\) into \(s\) sets \(U_1, \ldots, U_s\), so that both:
\(e(G[U_i]) \leq \frac{e(G)}{s^2} + \sqrt{\frac{e(G)}{s}}, \quad \text{for } i = 1, \ldots, s\) and \(\sum\limits_{i=1}^s e(G[U_i]) \leq \frac{e(G)}{s}.\)

The basic interpolation theorem states that if graph \(G\) contains spanning trees having \(m\) and \(n\) leaves, with \(m < n\), then for each integer \(k\) such that \(m < k < n\), \(G\) contains a spanning tree having \(k\) leaves. Various generalizations and related topics will be discussed.