19 research outputs found
The colourful simplicial depth conjecture
Given sets of points, or colours, in , a colourful simplex is a set such that
, for all . The colourful
Carath\'eodory theorem states that, if is in the convex hull of
each , then there exists a colourful simplex containing in
its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth,
Discrete Comput. Geom., 35, 597--604 (2006)) conjectured that, when
for all , there are always at least colourful
simplices containing in their convex hulls. We prove this
conjecture via a combinatorial approach
Tropical Carathéodory with Matroids
Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth
Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their
properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic
and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization,
Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions.
This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
Tverberg's theorem is 50 Years Old: A survey
This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. © 2018 American Mathematical Society