A family of sets is evolutionary if it is possible to order its elements such that each set except the first one has an element in the union of the previous sets and also an element not in that union. This definition is inspired by a conjecture of Naddef and Pulleyblank concerning ear decompositions of 1-extendable graphs. Here we consider the problem of determining whether a family of sets is ...

We consider the problem of minimizing the size of a family of sets G such that every subset of {1, . . . , n} can be written as a disjoint union of at most k members of G, where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the question of finding the minimum of |G| so that every subset of {1, ....

We study an ordered version of hypergraph Ramsey numbers using linearly ordered vertex sets, due to Fox, Pach, Sudakov, and Suk. In the k-uniform ordered path, the edges are the sets of k consecutive vertices in a linear vertex order. Moshkovitz and Shapira described its ordered Ramsey number as the solution to an enumeration problem involving higher-order mulidimensional integer partitions. We...

We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström analogous conjecture Kahn for independent sets in regular graphs hold all large enough as a function degree. That is, every d n divisible by 2d, union n/(2d) copies complete d-regular bipartite graph maximizes number matchings size k each over on vertices. To this we utilize cluster expansion canonical ensemble stat...

We consider the problem of minimizing the size of a family of sets G such that every subset of {1, . . . , n} can be written as a disjoint union of at most k members of G, where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the minimum of |G| so that every subset of {1, . . . , n} is the union o...

Some properties of minimal open sets and maximal open sets are studied in [1, 2]. In this paper, we define dual concepts of them, namely, maximal closed set and minimal closed set. These four types of subsets appear in finite spaces, for example. More generally, minimal open sets and maximal closed sets appear in locally finite spaces such as the digital line. Minimal closed sets and maximal op...

In this paper a new class of sets called  regular generalized  $delta$-closed set (briefly rg$delta$-closed set)is introduced and its properties are studied. Several examples are provided to illustrate the behaviour of these new class of sets.

Given a nite number of closed convex sets whose algebraic representation is known, we study the problem of optimizing a convex function over the closure of the convex hull of the union of these sets. We derive an algebraic characterization of the feasible region in a higher-dimensional space and propose a solution procedure akin to the interior-point approach for convex programming.

We show that under mild conditions, a Gaussian analytic function F that a.s. does not belong to a given weighted Bergman space or Bargmann–Fock space has the property that a.s. no non-zero function in that space vanishes where F does. This establishes a conjecture of Shapiro (1979) on Bergman spaces and allows us to resolve a question of Zhu (1993) on Bargmann–Fock spaces. We also give a simila...

In the PhD thesis by Burris (Memphis (1993)), a conjecture was made concerning the number of colors c(G) required to edge-color a simple graph G so that no two distinct vertices are incident to the same multiset of colors. We find the exact value of c(G) — the irregular coloring number, and hence verify the conjecture when G is a vertexdisjoint union of paths. We also investigate the point-dist...