We give some results on the excess of Hadamard matrices. We provide a list for Hadamard matrices of order \(\leq 1000\) of the smallest upper bounds known for the excess for each order. A construction is indicated for the maximal known excess.

The type of a \(3\)-factorization of \(3K_{2n}\) is the pair \((t,s)\), where \(t\) is the number of doubly repeated edges in \(3\)-factors, and \(\binom{n}{2} – s\) is the number of triply repeated edges in \(3\)-factors. We determine the spectrum of types of \(3\)-factorizations of \(3K_{2n}\), for all \(n \geq 6\); for each \(n \geq 6\), there are \(43\) pairs \((t,s)\) meeting numerical conditions which are not types and all others are types. These \(3\)-factorizations lead to threefold triple systems of different types.

Let \(V\) be a finite set of \(v\) elements. A covering of the pairs of \(V\) by \(k\)-subsets is a family \(F\) of \(k\)-subsets of \(V\), called blocks, such that every pair in \(V\) occurs in at least one member of \(F\). For fixed \(v\), and \(k\), the covering problem is to determine the number of blocks of any minimum (as opposed to minimal) covering. Denote the number of blocks in any such minimum covering by \(C(2,k,v)\). Let \(B(2,5,v) = \lceil v\lceil{(v-1)/4}\rceil/{5}\rceil\). In this paper, improved results for \(C(2,5,v)\) are provided for the case \(v \equiv 1\) \(\quad\) or \(\quad\) \(2 \;(mod\;{4})\).\(\quad\) For \(\quad\) \(v \equiv 2\; (mod\;{4})\), \(\quad\) it \(\quad\) is \(\quad\) shown \(\quad\) that \(C(2,5,270) = B(2,5,270)\) and \(C(2,5,274) = B(2,5,274)\), establishing the fact that if \(v \geq 6\) and \(v \equiv 2\;mod\;4\), then \(C(2,5,v) = B(2,5,v)\). In addition, it is shown that if \(v \equiv 13\;(mod\;{20})\), then \(C(2,5,v) = B(2,5,v)\) for all but \(15\) possible exceptions, and if \(v \equiv 17\;(mod\;{20})\), then \(C(2,5,v) = B(2,5,v)\) for all but \(17\) possible exceptions.

The structure and the hamiltonicity of vertex-transitive graphs of order \(qp\), where \(q\) and \(p\) are distinct primes, are studied. It is proved that if \(q < p\) and \(\text{p} \not\equiv 1 \pmod{\text{q}}\) and \(G\) is a vertex-transitive graph of order \(qp\) such that \({Aut}G\) contains an imprimitive subgroup, then either \(G\) is a circulant or \(V(G)\) partitions into \(p\) subsets of cardinality \(q\) such that there exists a perfect matching between any two of them. Partial results are obtained for \(\text{p} \equiv 1 \pmod{\text{q}}\). Moreover, it is proved that every connected vertex-transitive graph of order \(3p\) is hamiltonian.

In this paper, the algorithm developed in \([RK]\) for \(2\)-color Ramsey numbers is generalized to multi-colored Ramsey numbers. All the cyclic graphs yielding the lower bounds \(R(3,3,4) \geq 30\), \(R(3,3,5) \geq 45\), and \(R(3,4,4) \geq 55\) were obtained. The two last bounds are apparently new.

Consider combinations of \(k\) out of \(n\) items as represented by bit-strings of length \(n\) with exactly \(k\) ones. An algorithm for generating all such combinations so that successive bit-strings differ by the interchange of a single \(01\) or \(10\) pair exists only if \(n\) is even and \(k\) is odd (except for the trivial cases where \(k = n, n-1, 0, 1\)). This was shown by Eades, Hickey, and Read \([4]\) (and others) but no explicit algorithm was given. Later, Carkeet and Eades \([3]\) gave an inefficient, exponential storage implementation. Here, we present an implementation of the algorithm of \([4]\) that is constant average time, and uses linear storage.

The minimum cardinality of a pairwise balanced design on nineteen points is determined; a minimal design is exhibited containing \(13\) triples and \(22\) quadruples.

It is shown that the collection of all the \(\dbinom{10}{3}\) triples chosen from a set of ten points can be partitioned into ten mutually disjoint \(2-(9,3,1)\) designs in precisely \(77\) non-isomorphic ways.

A \((3,k,n,e)\) Ramsey graph is a triangle-free graph on \(n\) vertices with \(e\) edges and no independent set of size \(k\). Similarly, a \((3,k)\)-, \((3,k,n)\)-, or \((3,k,n,e)\)-graph is a \((3,k,n,e)\) Ramsey graph for some \(n\) and \(e\). In the first part of the paper, we derive an explicit formula for the minimum number of edges in any \((3,k,n)\)-graph for \(n\leq3(k-1)\), i.e., a partial formula for the function \(e(3,k,n)\) investigated in \([3,5,7]\). We prove some general properties of minimum \((3,k,n)\)-graphs with \(e(3,k,n)\) edges and present a construction of minimum \((3,k+1,3k-1,5k-5)\)-graphs for \(k\geq2\) and minimum \((3,k+1,3k,5k)\)-graphs for \(k\geq4\). In the second part of the paper, we describe a catalogue of small Ramsey graphs: all \((3,k)\)-graphs for \(k\leq6\) and some \((3,7)\)-graphs, including all \(191 (3,7,22)\)-graphs, produced by a computer. We present for \(k\leq7\) all minimum \((3,k,n)\)-graphs and all \(10\) maximum \((3,7,22)\)-graphs with \(66\) edges.