Characterising quotients of projective Fraïssé limits
This nice paper studies projective Fraïssé limits of families of finite graphs and their quotients. The notion of projective Fraïssé family of topological structures and its limit was introduced by T. Irwin and S. Solecki; the main idea is dualising the usual model theoretic construction of a Fraïssé limit. They showed that the continuum known as the pseudo-arc can be obtained as the quotient of the limit of a family of reflexive linear graphs. This paper goes deeper in the analysis showing that there are few spaces which can be obtained as quotients of limits of projective Fraïssé limits of families of reflexive finite graphs. The idea is that given a family of graphs \((G,R)\), its limit is a space \((D,R^D)\) where \(R^D\) ends up to be a equivalence relation on \(D\). The quotient can only be: {\parindent5mm \begin{itemize}\item[1)] a Cantor space, \item[2)] a disjoint sum of \(m\) singletons and \(n\) pseudo-arcs, for \(m,n\in\mathbb{N}\) and \(m+n>0\), \item[3)] a disjoint sum of \(n\) compact metric spaces \(X_i\), each consisting of a disjoint union \(P_i\cup\bigcup_{j\in\mathbb{N}} Q_{i,j}\) where: \end{itemize}} {\parindent4mm \begin{itemize}\item[--] \(P_i\) is a pseudo-arc, \item[--] each \(Q_{i,j}\) is a Cantor space clopen in \(X_i\), \item[--] \(\bigcup_{j\in\mathbb{N}}Q_{i,j}\) is dense in \(X_i\). \end{itemize}} A key result in the paper is the one stating that if \(\Delta\) is a projective Fraïssé family of finite connected graphs of unbounded diameters, then \(\Delta\) is contained in the family of finite linear graphs. In other words, except when the diameter of its elements is bounded or it contains only linear graphs, any projective Fraïssé subfamily of graphs contains disconnected graphs and hence the quotient of the limit cannot be connected.
scientific article; zbMATH DE number 5724593