A Constructive Proof of Ky Fan's Generalization of Tucker's Lemma

The Borsuk-Ulam-property, Tucker-property and constructive proofs in combinatorics

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 introduced and its relation to the topological Borsuk-Ulam-property is discussed. Applications of the Tucker-property in combinatorics are demonstrated.

Generalizations of Tucker-Fan-Shashkin lemmas

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 these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about the doubling of manifolds with boundaries that are BUT-manifolds.

球丛的 Ky Fan 定理

Ky Fan theorem for sphere bundles

本文将经典 Ky Fan 定理从三角剖分球面推广到三角剖分球丛的全空间，扩展了关于对跖着色的组合拓扑结论。

A Geometric Approach to Combinatorial Fixed-Point Theorems

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 doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological f…

Colorful subgraphs in Kneser-like graphs

Combining Ky Fan's theorem with ideas of Greene and Matousek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.