For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x, y\in X\text{ and } x\neq y\big\}$. 


Let $G=(V,E)$ be a simple, undirected graph, and suppose ${\cal V}$ is a collection of subsets of $V$ such that for all $W, W' \in {\cal V}$ we have either $W\subseteq W'$ or $W' \subseteq W$.


If for all $W\in {\cal V}$ the induced subgraph $(W,E\cap [W]^2)$ is [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph), is the induced subgraph $(\bigcup {\cal V}, E\cap [\bigcup{\cal V}]^2)$ necessarily vertex-transitive as well?


(Applying [Zorn's Lemma](https://en.wikipedia.org/wiki/Zorn%27s_lemma), which is equivalent to the [Axiom of Choice](https://en.wikipedia.org/wiki/Axiom_of_choice), a positive answer would imply that every graph has a maximal vertex-transitive subgraph.)



Ascending chain of vertex-transitive graphs
Let $Q$ be a chain (totally ordered set). Let $Y\_Q$ be the graph whose set of vertices is the set of pairs $(x,y)$ with $x,y\in Q$ and $x<y$, and an edge between $(x,y)$ and $(x',y')$ iff the intervals $]x,y[$ and $]x',y'[$ are disjoint. 


If $Q$ is a 2-transitive chain then $Y\_Q$ is a vertex-transitive graph.


Let $D$ be an "open right" long line, so $D$ is locally order-isomorphic to the reals, for every $x$, the set $\{y\in D:y<x\}$ is order-isomorphic to $\mathbf{R}$ and every countable subset has an upper bound;


Let $X$ be obtained by concatenation $D\sqcup\{x\_0\}\sqcup\mathbf{R}$ of a copy of $D$, a singleton $x\_0$, and a copy of $\mathbf{R}$.


I claim that $Y\_X$ is not vertex-transitive. Indeed, if $x<x\_0<y$, then any complete subgraph in $Y\_X$ containing the vertex $(x,y)$ is countable. On the other hand, there is an uncountable complete subgraph (consider uncountably may disjoint nonempty open intervals in $D$).


Write $D=\bigcup D\_\alpha$, ascending union of a chain of intervals all order-isomorphic to $\mathbf{R}$, and $X\_\alpha=D\_\alpha\cup\{x\_0\}\cup\mathbf{R}$; then $X\_\alpha$ is order-isomorphic to $\mathbf{R}$. Then $X=\bigcup X\_\alpha$, and $Y\_X=\bigcup Y\_{X\_\alpha}$, union of a chain of vertex-transitive subgraphs.


