Elementary proof techniques for the maximum number of islands

Abstract

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.