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Structural convergence and algebraic roots
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 23 December 2024, pp. 1-9
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- Article
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Structural convergence is a framework for the convergence of graphs by Nešetřil and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs
$(G_n)$ converging to a limit 
$L$ and a vertex 
$r$ of 
$L$, it is possible to find a sequence of vertices 
$(r_n)$ such that 
$L$ rooted at 
$r$ is the limit of the graphs 
$G_n$ rooted at 
$r_n$. A counterexample was found by Christofides and Král’, but they showed that the statement holds for almost all vertices 
$r$ of 
$L$. We offer another perspective on the original problem by considering the size of definable sets to which the root 
$r$ belongs. We prove that if 
$r$ is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots 
$(r_n)$ always exists.
