# Colorful phenomena in discrete geometry and

combinatorics via topological methods

## Shira Zerbib, University of Michigan, Ann Arbor

## Date and time: 2pm, Friday, November 30, 2018

## Place: CUNY Graduate Center, Rm 4419

We will discuss two new topological theorems and their applications to
different problems in discrete geometry and combinatorics involving
colorful settings.

The first result is a polytopal-colorful generalization of the
topological KKMS theorem due to Shapley. We apply our theorem to prove
a new colorful extension of the well-known $d$-interval theorem of
Tardos and Kaiser, as well as to provide a new proof to the colorful
Caratheodory theorem of Bárány. Our theorem can be also applied to
questions regarding fair-division of goods among a set of
players. This is a joint work with Florian Frick.

The second result is a new topological lemma that is reminiscent of
Sperner's lemma: instead of restricting the labels that can appear on
each face of the simplex, our lemma considers labelings that enjoy a
certain symmetry on the boundary of the simplex. We apply this to
prove that the well-known envy-free division theorem of a cake is true
even if the players are not assumed to prefer non-empty pieces (that
is, without the "hungry players" condition), if the number of players
is prime or equal to 4. This is joint with Frederic Meunier.