Tightness problems in the plane

Tightness problems in the plane

A 3-uniform hypergraph is called tight if for any 3-coloring of its vertex set a heterochromatic edge can be found. In this paper we study tightness of 3-graphs with vertex set IR and edge sets arising from simple geometrical considerations. Basically we show that sets of triangles with \fat shadows" are tight and also that some interesting sets of triangles with \thin shadows" are tight.

Tightness of Constraint Satisfaction Problems

We define a new measure of constraint tightness. This can be used to compare constraint satisfaction problems with different numbers of constraints and constraints of different arities. This measure possesses a number of desirable properties. For example, it is a partial order and forms a lattice over the set of constraint satisfaction problems. We illustrate the value of this measure by provin...

Constrained Pursuit - Evasion Problems in the Plane

In pursuit-evasion problems, we are presented with one or more pursuers attempting to capture one or more evaders. We consider pursuers and evaders limited by a maximum speed moving in the two-dimensional plane with obstacles. We then investigate two problems in this domain. In the first, where we are given the starting configuration of pursuers and evaders, we identify all possible paths by th...

Some Combinatorial Problems in the Plane

1 Let there be given n points in the plane. Denote by t i the number of lines which contain exactly i of the points (2 < i < n). The properties of the set {ti } have been studied a great deal. For example, there is the classical result of Gallai and Sylvester : Assume to = 0 (i .e ., the points are not all on one line) ; then t2 > 0. For the history of this problem see, e.g ., Motzkin [6] and E...

Some Separability Problems in the Plane