Let G and H be graphs, with |G| ≤ |H|. Consider a one-to-one map f:G→H, and define |f| = min{distH(f(x),f(y)): xy E(G)}. Now let the separation of G in H be defined by sep(G,H) = max{|f|: all one-to-one maps f:G→H}. We survey some results of the past several years on this parameter, in particular using Alon and Milman's results on the connection between the second largest eigenvalue of a graph ...

We present examples of flag homology spheres whose γvectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In another direction, we show that if a flag (d− 1)-sphere has at most 2d + 2 vertices its γ-vector satisfies the Kruska...

BERNARDO M. ÁBREGO, SILVIA FERNÁNDEZ-MERCHANT, AND BERNARDO LLANO Abstract. Given integers k ≥ 1 and n ≥ 0, there is a unique way of writing n as n = ¡nk k ¢ + ¡ nk−1 k−1 ¢ + ... + ¡ n1 1 ¢ so that 0 ≤ n1 < · · · < nk−1 < nk. Using this representation, the KruskalMacaulay function of n is defined as mk (n) = ¡ nk−1 k−1 ¢ + ¡ nk−1−1 k−2 ¢ + ... + ¡ n1−1 0 ¢ . We show that if a ≥ 0 and a < mk+1 (...

We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Fried...

The energy content of electrostatic perturbations about homogeneous equilibria is discussed. The calculation leading to the well-known dielectric (or as it is sometimes called, the wave) energy is revisited and interpreted in light of Vlasov theory. It is argued that this quantity is deficient because resonant particles are not correctly handled. A linear integral transform is presented that so...

The aim of this paper is to examine the interdependence between adult education and competitiveness innovative potential economy in conditions sustainable development. To mentioned, we used a sample data that includes European countries. Global Competitiveness Index its components were taken for research as numerical display country’s competitiveness. For indicators lifelong learning, share per...