Volume 80, Nº 3 (2025)

We consider questions relating to the convergence of generalized power series (with complex-valued exponents) that satisfy formally some analytic functional equations: a differential equation, a $q$-difference one, or Mahler's equation. We present new results, as well as generalizations of some of our earlier results, thus summing up our investigations of this subject. We also present a selection of results on the existence and uniqueness of local holomorphic solutions of such equations and review some classical results on the convergence of Taylor power series that solve them formally.

One of the most important objects of study in modern graph theory is the so-called Johnson graph $G(n,r,s)$. The vertices of this graph are all $C_n^r$ subsets of cardinality $r$ of the set $\{1,…,n\}$, and two vertices are joined by an edge if and only if the intersection of the corresponding subsets has cardinality $s$. Such graphs play a vital role in coding theory, extremal combinatorics and Ramsey theory, combinatorial geometry, and in other areas. In this survey we give an account of applications of these graphs and of some of their parameters including the independence number, the clique number, and the chromatic number. We also pay attention to a large part of this theory concerned with the investigation of random subgraphs of Johnson graphs, which was actively developed in recent years.