A conjecture of Grünbaum on 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 ...

Common transversals for partitions of a finite set

For s 2, t 1, let A1; : : : ;At be s-cell partitions of a finite set X . Assume that if x;y 2 X , x 6= y, then x;y belong to different cells of at least one of the partitions Ai. For each k 1, let c(s; t;k) be the least integer such that if A1; : : : ;At , X satisfy the preceding conditions, and the smallest of all the cells of all the partitions has exactly k elements, and jX j c(s; t;k), then...

Tverberg's Transversal Conjecture and Analogues of Nonembeddability Theorems for Transversals

In this paper we prove a special case of the transversal conjecture of Tverberg and Vrećica. We consider the case when the numbers of parts ri in this conjecture are powers of the same prime. We also prove some results on common transversals that generalize the classical nonembeddability theorems. We also prove an analogue of colored Tverberg’s theorem by Živaljević and Vrećica. Instead of mult...