For r = 7, 8 we prove that if χ(G) = r, then cr(G) ≥ cr(Kr). The case r = 5 of the Weak Hajós Conjecture is equivalent to the Four Color Theorem, and the case r = 6 follows from a theorem of Oporowski and Zhao. We present partial results when r = 9.

This paper presents an experiment with human subjects concerning the vague spatial relation ‘near’ in environmental space. After the topic is introduced and relevant previous work surveyed, the experiment is described. Three approaches to experimental analysis are presented and discussed: nearness neighbourhoods as regions with broad boundaries, fuzzy nearness and distance measures, and four-va...

Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the formulation of a “one-leaf” analogue of the classical Frobenius integrability theorem in elementary differential geometry. Several applications of the result will...

We present a statement about Lie algebras that is equivalent to the Four Color Theorem.

Nearness (a fuzzy nearness) is a fuzzy relation that can be used to model various grades of “being close” in a linear space. We study the uniform convergence of a sequence of functions with values in a space equipped with a nearness relation. The uniform convergence for the mappings into a space with a fuzzy nearness is defined and it is shown that a theorem similar to Moore-Osgood theorem for ...

We look at the proofs of two fundamental theorems in Ramsey theory, Van der Waerden’s Theorem and the Hales-Jewett Theorem. In addition, we study bounds on Van der Waerden numbers.

A one-dimensional spatial model of electoral competition with regional parties is considered, where parties only want to win constituencies. With no regional parties and only two national parties, the median voter theorem holds. But when we consider one national party and one regional party for each of the two regions, then the median voter theorem breaks down. Equilibrium is characterized unde...

Morley's trisector theorem states that " The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle " [10]. The notation and terminology used in this paper have been introduced in the

A study of 7735 middle-aged British men drawn from general practices in twenty-four towns shows that there has been a progressive increase in mean height in the men who were born between 1919 and 1939. This is true for both manual and non-manual classes, but the mean heights of the two groups are significantly different and remain widely separated over this period of time. Manual workers lag tw...