7 papers · 1 filter
The topological Bárány-Larman conjecture for prime numbers
Pablo Soberón
The Bárány-Larman conjecture states that for any sets of points each in , considered as color classes, we can partition their union into rainbow …
Helly and Radon theorems for convex intersections containing -flats
Sarah Ludwigson, Pablo Soberón
We study versions of results in combinatorial geometry related to families of convex sets in whose intersection contains a -dimensional affine space. We prove gen…
Four hyperplanes do not always equipartition a mass in
Pablo Soberón
We construct a smooth strictly positive density in that cannot be divided into parts of the same size by four affine hyperplanes. This settles the last open cas…
Blocking codimension-one simplices on the moment curve
Pablo Soberón
We study , the minimum number of points needed to meet the relative interior of every -simplex spanned by an -point set in general position in . In…
Selection-structure generalizations of the Borsuk-Ulam theorem
Pablo Soberón
We prove Borsuk-Ulam-type results governed by selection structures. Selection structures extend the matroidal framework for colorful theorems in discrete geometry and include non-m…
KKM theorems and discrete geometry beyond matroids
Pablo Soberón
We introduce selection structures, a topological framework that extends the role played by color classes and matroids in discrete geometry and KKM theorems. Selection structures al…
Tverberg cores and Kalai's cascade conjecture
Pablo Soberón
We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an -simplex to , let be the set of points contained in the images…