Given a finite graph H and G, a subgraph of it, we define σ (G, H) to be the largest integer such that every pair of subgraphs of H, both isomorphic to G, has at least σ(G, H) edges in common; furthermore, R(G, H) is defined to be the maximum number of subgraphs of H, all isomorphic to G, such that any two of them have σ(G, H) edges common between them. We are interested in the values of σ(G, H) and R(G, H) for general H and G. A number of combinatorial problems can be considered as special cases of this question; for example, the classical set-packing problem is equivalent to evaluating R (G, H) where G is a complete subgraph of the complete graph H and σ(G, H) = 0, and the decomposition of H into subgraphs isomorphic to G is equivalent to showing that σ(G, H) = 0 and R(G, H) = ε(H)/ε(G) where ε(H), ε(G) are the number of edges in H, G respectively.

A result of S. M. Johnson (1962) gives an upper bound for R(G, H) in terms of σ(G, H). As a corollary of Johnson's result, we obtain the upper bound of McCarthy and van Rees (1977) for the Cordes problem. The remainder of the paper is a study of σ (G, H) and R(G, H) for special classes of graphs; in particular, H is a complete graph and G is, in most instances, a union of disjoint complete subgraphs.

On page 354 of James (1975) (in the proof of Theorem 5.3(b)) there appear the following two sentences:

‘When n≥7, replacewe may suppose δ=0.’

Unfortunately, the second sentence does not follow from the first. In fact, the first sentence also forces us to replace remains unaltered. Thus the case δ=1 is omitted in the paper.

We identify a large class of rings over which locally free modules are determined by their endomorphism rings. We characterize these endomorphism rings and consider under what circumstances the conditions on the locally free modules can be relaxed, for example by requiring that only one of the rings need be in the special class, or by replacing ‘free' by “projective”.

By using pairwise relatively complete families of functions, which we define here, we obtain two characterizations of quasi-pseudometrizable bitopological spaces from which some known theorems can be derived as easy corollaries.

If the second dual of a Banach space E is smooth at each point of a certain norm dense subset, then its first dual admits a long sequence of norm one projections, and these projections have ranges which are suitable for a transfinite induction argument. This leads to the construction of an equivalent locally uniformly rotund norm and a Markuschevich basis for E*.

This note establishes estimates for lower bounds of linear forms at algebraic points of E-functions and G-functions defined over a general algebraic number field.

A double triangle subspace lattice in a Hilbert space H is a 5-element set of subspaces of H, containing (0) and H, with each pair of non-trivial elements intersecting in (0) and spanning H. It is shown that if any pair of non-trivial elements has a closed vector sum the double triangle is both non-reflexive and non-transitive. A double triangle in H⊕H is an operator double triangle if each non-trivial elements is the graph of an operator acting on H. A sufficient condition is given for any operator double triangle to be non-reflexive.

We have defined a mini-injective module and given some structures of self mini-injective rings and certain relationships between such rings and QF-rings in [8] and [9].

In this short note we shall study the modules dual to mini-injective modules, which we call maxi-quasiprojective modules. We shall give a characterization and some structures, in terms of the above modules, of those rings whose every injective modules has the lifting property of direct decompositions modulo the Jacobian radical (see [5], [6] and [7]). Furthermore, we shall show that the above rings are closely related to QF-rings (see [8] and [9]).

A maximal tolerance of a lattice L without infinite chains is either a congruence or a central relation. A finite lattice L is order-polynomially complete if and only if L is simple and has no central relation.

All inverse semigroups with idempotents dually well-ordered may be constructed inductively. The techniques involved are the constructions of ordinal sums, direct limits and Bruck-Reilly extensions.

If two functions of a real variable are integrable over two intervals, say of t, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.

It is proved that an arbitrary pair of positive integers can be simultaneously represented by products of the values at integer points of certain rational functions. Linear recurrences in Z-modules and elliptic power sums are applied.