Maximal 2-distance sets containing the regular simplex

New Maximal Two-Distance Sets

A two-distance set in E is a point set X in the d-dimensional Euclidean space such that the distances between distinct points in X assume only two different nonzero values. Based on results from classical distance geometry, we develop an algorithm to classify, for a given d, all maximal (largest possible) two-distance sets in E. Using this algorithm we have completed the full classification for...

Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let n > 3 be a natural number. We study the problem of finding the smallest r such that there is a family A of 2-subsets and 3-subsets of [n] = {1, 2, . . . , n} with the following properties: (1) A is an antichain, i.e., no member of A is a subset of any other member of A, (2) A is maximal, i.e., for every X ∈ 2 \ A there is an A ∈ A with X ⊆ A or A ⊆ X, and (3) A is r-regular, i.e., every poi...

On Inscribing «-dimensional Sets in a Regular »-simplex

Some results on maximal open sets

In this paper, the notion of maximal m-open set is introduced and itsproperties are investigated. Some results about existence of maximal m-open setsare given. Moreover, the relations between maximal m-open sets in an m-spaceand maximal open sets in the corresponding generated topology are considered.Our results are supported by examples and counterexamples.

Counting maximal distance-independent sets in grid graphs

Previous work on counting maximal independent sets for paths and certain 2-dimensional grids is extended in two directions: 3-dimensional grid graphs are included and, for some/any l ∈ N, maximal distance-l independent (or simply: maximal l-independent) sets are counted for some grids. The transfer matrix method has been adapted and successfully applied.