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.