Symmetric twins and common transversals

Common Transversals

Given t families, each family consisting of s finite sets, we show that if the families “separate points" in a natural way, and if the union of all the sets in all the families contains more than (s+1)t st 1 1 elements, then a common transversal of the t families exists. In case each family is a covering family, the bound is st st 1. Both of these bounds are best possible. This work extends rec...

Geometric Permutations and Common Transversals

Efficient Algorithms for Common Transversals

Suppose we are gIven a set S of n (possibly intersecting) simple objects in the plane, such that for every pair of objects in S, the intersection of the boundaries of these two objects has at most a connected components. The integer a is independent of n, Le. a.=O (1). \Ve consider the problem of detennining whether there exists a straight line that goes through every object in S. We give an 0 ...

Disjoint Common Transversals and Exchange Structures

We state and prove a theorem (Theorem 1 below) which strengthens previously known results concerning disjoint common partial transversals of two families of sets. This theorem may be viewed as a result on transversal pre-independence structures. We define a "disjoint-exchange structure" on a set and extend the result to such structures (Theorem 3 below). Then we give an application of this theo...

Theorems of Borsuk-ulam Type for Flats and Common Transversals

In this paper some results on the topology of the space of k-flats in Rn are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in Rn, and on measure partitions by hyperplanes, are deduced.