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 (...