Elementary proof techniques for the maximum number of islands

TL;DR

This paper introduces elementary proof techniques using rooted binary trees and discrete geometry to analyze extremal properties of islands on various combinatorial boards, unifying previous approaches and extending known results.

Contribution

It presents a unified elementary framework for analyzing maximum islands, including new bounds on toroidal and hypercube boards, and refines neighborhood relations.

Findings

Maximum number of islands on a toroidal board determined

Maximum islands in a hypercube established

Neighborhood relation strength is improved

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.