Islands: from coding theory to enumerative combinatorics and to lattice theory - overview and open problems

Abstract

As a tool to characterize instantaneous codes, Foldes and Singhi determined the maximum number of certain subsets of the set of the first n natural numbers in 2006. This motivated Czédli in 2009 to determine the maximum number of the analogous subsets of a rectangular grid. He called these subsets islands. The present paper summarizes recent results on this topic.

CD-independent subsets in meet-distributive lattices

“…By giving a new proof for Proposition 4.1 based on CD-independence, the goal of this section is to demonstrate that Lattice Theory is still competitive with other approaches. Note that, besides that this was the original motivation in Czédli, Hartmann and Schmidt [12], this task was also suggested by Horváth [24,Problem 9.1]. We only need Proposition 1.3, taken from [12], for this purpose.…”

“…This result was soon followed by many related ones, due to Barát, Foldes, E. K. Horváth, G. Horváth, Lengvárszky, Németh, Pach, Pluhár, Pongrácz, Šešelja, Szabó, and Tepavčević. The results of these authors, written alone or in various groups, range from triangular boards to the continuous case and from lattice theory to combinatorics, see [6], [21], [24], [25], [26], [28], [29], [30], [32], [33], and some further papers not referenced here. Since [21] and [24] give good overviews on islands, we do not go into further historical details.…”

“…The results of these authors, written alone or in various groups, range from triangular boards to the continuous case and from lattice theory to combinatorics, see [6], [21], [24], [25], [26], [28], [29], [30], [32], [33], and some further papers not referenced here. Since [21] and [24] give good overviews on islands, we do not go into further historical details. However, we mention the following feature of this research field.…”