4 research outputs found
Colorful Borsuk--Ulam theorems and applications
We prove a colorful generalization of the Borsuk--Ulam theorem and derive
colorful consequences from it, such as a colorful generalization of the ham
sandwich theorem. Even in the uncolored case this specializes to a
strengthening of the ham sandwich theorem, which given an additional condition,
contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated
measures as a special case. We prove a colorful generalization of Fan's
antipodal sphere covering theorem, we derive a short proof of Gale's colorful
KKM theorem, and we prove a colorful generalization of Brouwer's fixed point
theorem. Our results also provide an alternative between Radon-type
intersection results and KKM-type covering results. Finally, we prove colorful
Borsuk--Ulam theorems for higher symmetry.Comment: 15 page
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