The possible number of islands on the sea

Abstract

Let a height function f be a real valued function on R 2 . A connected subset of R 2 is called an island, if there is a water level such that H is an island in the classical sense. We show that an island system is always laminar. Among others, in this paper we prove that the cardinality of a maximal laminar system is either countable or continuum.

Cardinality of height function’s range in case of maximally many rectangular islands — computed by cuts

“…Horváth, Németh and Pluhár determined upper and lower bounds for the maximum number of triangular islands on a triangular grid in [9]. Some further interesting investigations and nice results on islands appear in [12,14,17]. The number of square islands is a similar problem to the triangular case and it is treated in [7,13].…”

Systems of Islands With Continuous Height functions

“…For discrete islands, being a system of islands is in essence equivalent to laminarity. In general, this is not true as shown by Pach et al [13]. They show, however, that for bounded connected sets in R n , a countable maximal laminar system H is also a (maximal) system of islands if and only if the distance of any two disjoint sets in H is positive.…”

“…The general concept of systems of islands by Pach et al [13] includes all these discrete island systems as well. For example, Czédli's rectangular islands can be obtained by letting Ω be the m by n rectangle with vertices at (0, 0), (m, 0), (m, n), and (0, n) in the plane and by letting S be the set of rectangles with sides parallel to the coordinate axes and vertices of integer coordinates.…”

“…Pach et al [13] consider laminar systems as well. A collection H of sets is called laminar if for every pair of sets H 1 , H 2 ∈ H, either H 1 ⊆ H 2 or H 2 ⊆ H 1 or H 1 ∩ H 2 = ∅.…”

“…A collection H of sets is called laminar if for every pair of sets H 1 , H 2 ∈ H, either H 1 ⊆ H 2 or H 2 ⊆ H 1 or H 1 ∩ H 2 = ∅. A system of islands that are connected sets is always laminar (see [13]). For discrete islands, being a system of islands is in essence equivalent to laminarity.…”