2 research outputs found
Topological methods in zero-sum Ramsey theory
A cornerstone result of Erd\H os, Ginzburg, and Ziv (EGZ) states that any
sequence of elements in contains a zero-sum subsequence
of length . While algebraic techniques have predominated in deriving many
deep generalizations of this theorem over the past sixty years, here we
introduce topological approaches to zero-sum problems which have proven
fruitful in other combinatorial contexts. Our main result (1) is a topological
criterion for determining when any -coloring of an -uniform
hypergraph contains a zero-sum hyperedge. In addition to applications for
Kneser hypergraphs, for complete hypergraphs our methods recover Olson's
generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we
(2) give a fractional generalization of the EGZ theorem with applications to
balanced set families and (3) provide a constrained EGZ theorem which imposes
combinatorial restrictions on zero-sum sequences in the original result.Comment: 18 page
Cooperative conditions for the existence of rainbow matchings
Let , and let be a family of non-empty sets of
edges in a bipartite graph. If the union of every members of
contains a matching of size , then there exists an -rainbow
matching of size . Upon replacing by , the result can be
proved both topologically and by a relatively simple combinatorial argument.
The main effort is in gaining the last , which makes the result sharp