最大岛数的初等证明技术
Elementary proof techniques for the maximum number of islands
摘要：
岛屿是组合对象，可以直观地定义在由有限数量的单元组成的板上。基于细胞的相邻关系，两个岛是包含的还是不相交的是一个基本性质。最近，许多极值问题已经用不同的方法得到了回答。我们显示基本的技术统一这些方法。我们的建筑部分是基于有根二叉树和离散几何。除其他外，我们显示了最大基数的岛屿上的环形板，并在一个超立方体。我们还加强了以前的结果，稀疏的邻里关系。
英文摘要：
Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. Based on the neighbor relation of the cells, it is a fundamental property that two islands are either containing or disjoint. Recently, numerous extremal questions have been answered using different methods. We show elementary techniques unifying these approaches. Our building parts are based on rooted binary trees and discrete geometry. Among other things, we show the maximum cardinality of islands on a toroidal board and in a hypercube. We also strengthen a previous result by rarefying the neighborhood relation.
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