Related papers: A Constructive Proof of Ky Fan's Generalization of…

The classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker's lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of…

We give an intuitive combinatorial proof of Ky Fan's covering lemma based on the Borsuk-Ulam theorem. We then show how this approach can be generalized to Ky Fan's covering lemma for several linear orders.

We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In…

We prove multiple generalizations of Fan's combinatorial labeling result for sphere triangulations. This can be seen as a comprehensive extension of the Borsuk--Ulam theorem. In typical applications, the Borsuk--Ulam theorem gives…

Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of…

The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk - Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that…

Pachner proved that all closed combinatorially equivalent combinatorial manifolds can be transformed into each other by a finite sequence of bistellar moves. We prove an analogue of Pachner's theorem for combinatorial manifolds with a free…

Sperner's lemma is a statement about labeled triangulations of a simplex. McLennan and Tourky (2007) provided a novel proof of Sperner's Lemma by examining volumes of simplices in a triangulation under time-linear simplex-linear…

We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner's coloring. It is also…

We prove that a jointly conservative family of geometric functors between rigidly-compactly generated tensor triangulated categories induces a surjective map on Balmer spectra. From this we deduce a fiberwise criterion for Balmer's…

We develop the technique of geometric realizations with algebraically independent (over the field of real algebraic numbers) coordinates of vertices and combine it with the oriented volume method inspired by work of McLennan and Tourky on…

Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the case n=2 was proposed by Tucker in 1945. Numerous generalizations and applications of the Lemma have appeared since then. In 2006 Meunier proved the Lemma in its…

We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized…

This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tucker-property of a finite group $G$ is…

The multiplicative fragment of Linear Logic is the formal system in this family with the best understood proof theory, and the categorical models which best capture this theory are the fully complete ones. We demonstrate how the Hyland-Tan…

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of…

Triangulations and higher triangulations axiomatize the calculus of derived cokernels when applied to strings of composable morphisms. While there are no cubical versions of (higher) triangulations, in this paper we use coherent diagrams to…

We present some results on equivariant KK-theory in the context of tensor triangular geometry. More specifically, for G a finite group, we show that the spectrum of the tensor triangulated subcategory of KK^G generated by the tensor unit…

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary…

In this paper we prove a combinatorial theorem for finite labellings of trees, and show that it is equivalent to a theorem for finite covers of metric trees and a fixed point theorem on metric trees. We trace how these connections mimic the…