论文标题
围巾的定理，简单和定向的矩形
Scarf's theorems, simplices, and oriented matroids
论文作者
论文摘要
1967年，赫伯特·围巾提出了基于Sperner的引理令人惊讶的类似物的Brouwer固定点定理的新证明。这种类似物是出于围巾在游戏理论和数学经济学方面的工作的动机。此外，围巾证明了Sperner的引理涉及向量的颜色的一般版本。本文以作者论文的精神Arxiv的精神来重新审视围巾的思想：1909.00940和Arxiv：2012.13104。之后，我们获得了论文的主要新结果，即，围巾结果的概括为颜色属于方向的矩阵。不假定对定向矩形理论的了解。在最后一节中，我们返回原始的围巾定理，并使用更古典的方法对欧几里得空间的组合拓扑进行谴责。另外，我们概括了Kannai定理。
In 1967 Herbert Scarf suggested a new proof of Brouwer fixed point theorem based on a surprising analogue of Sperner's lemma. This analogue was motivated by Scarf's work in game theory and mathematical economics. Moreover, Scarf proved a much general version of Sperner's lemma dealing with colorings by vectors. The present paper begins by revisiting Scarf's ideas from the point of view of the basic theory of simplicial cochains in the spirit of author's papers arXiv:1909.00940 and arXiv:2012.13104. After this we get to the main new results of the paper, namely, to a generalization of Scarf results to colorings with colors belonging to an oriented matroid. No knowledge of the theory of oriented matroids is assumed. In the last section we return to the original Scarf theorem and reprove it using even more classical methods of the combinatorial topology of Euclidean spaces. Also, we generalize a theorem of Kannai.
