L. Moser [3] recently gave a very simple proof that

1.

has no solutions. In the present note we shall first of all prove that for , which by the fact that there is a prime p satisfying n < p ≤ 2n immediately implies that

2.

has no solutions. It is easy to see on the other hand that

3.

has infinitely many non-trivial solutions.

Let g(k) denote the least integer such that every graph , with n vertices and n+g(k) edges, contains at least k edge-disjoint cycles; let h(k) be similarly defined for planar graphs. Loops and multiple edges (i.e., cycles of length one and two) are permitted in both cases.

A real matrix A is said to be non-negative if and only if none of its entries is negative. Suppose A is an r by r non-negative matrix. We want to examine:

1. (A) The first power of A to maximize the number of positive entries in An,

2. (B) For each 1 ≤ i ≤ r the first power of A to maximize the number of positive entries in the i-th row of An.

We shall call the former first power the index of A and the latter the i-th row index of A (index (i, A)).

One of the most interesting of the smaller BIBD's is the system (8, 14, 7, 4, 3), where we write the parameters in the standard order v, b, r, k, λ. One representation of a design with these parameters is 1248, 3567; 2358, 1467; 3468, 1257; 4578, 1236; 5618,2347; 6728,1345; 7138,2456. This particular design has the feature that every block B is paired with a complementary block B' consisting of all varieties not lying in B. Thus

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:

If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.

In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).

Melzak [2] has shown that there exists a convex pseudopolyhedron Q (the convex hull of a convergent sequence of points together with its limit point) in E3 which is s-universal for triangles, that is, all possible triangles occur (up to similarity) as plane sections of Q. He conjectured that no polyhedron P has this property. In this short note we give an elementary proof of this conjecture.

A system of circles is said to form a k-fold packing of the plane if every point of the plane is an interior point of at most k circles. In this paper we consider only lattice packings, that is, we suppose that the centers of the circles form a lattice. We further restrict our consideration to circles of equal radius. Without loss of generality, the circles may have unit radius.

Let be a polynomial of degree n. Then clearly

(1.1).

(1.2).

and for R > 1

(1.3).

Note that if w = p(z) maps |z|<1 on a domain D of the w-plane then the area of D is given by

This expository note consists of two parts: In the first we present a homological formulation of Goursat's theorem. In the second we indicate how some rudimentary concepts of homological algebra can be introduced with the help of this form of Goursat's theorem. The second part is addressed to those readers with an algebraic background who wish to be initiated into homological algebra as painlessly as possible. The ideas developed here are close to the spirit of [5, Chapter II, § 6], where further references may be found. In fact, these ideas are very much in the air, and any originality in the present note is purely coincidental.

A category with zero-maps is called "quasi-exact" in the sense of D. Puppe (see [4], page 8, 2. 4), if it satisfies the following axioms:

1. (Q1) Every may f is a product f=με of an epimorphisrn εfollowed by a monomorphism μ

1. (Q2) a) Every epimorphism ε has a kernel k = ker ε

b) Every monomorphism μ has a cokernel γ = Coker ε, where Ker and Coker are characterized by the familiar universality properties (see [3], page 252, (1. 10) and (1. 11)).

This note is based on Hu's Homotopy Theory (New York, 1959), pp. 115 f.

The theorem that the homotopy sequence is exact splits into six statements. The fifth in Hu's exposition can be stated as follows: Let f:In, In-1, Jn-1→X, A, x0,

1

Then there exists a map

2

such that f~j g: In, In-1, Jn-1→X, A, x0.

A special case of a well known theorem of Ramsay [3] states that an infinite graph either contains an infinite complete subgraph or it contains an infinite independent set; in other words there exists an infinite subset of its vertices so that either every two of them are joined by an edge or no two of them are joined by an edge. Thus if we have a graph whose vertices are the integers, and which has no infinite complete sub-graph, it certainly has an infinite independent set. The question can now be asked if there exists an independent set whose vertices n1 < n2 < … do not tend to infinity too fast.