Modeling Limits in Hereditary Classes: Reduction and Application to Trees

Abstract

The study of limits of graphs led to elegant limit structures for sparse and dense graphs. This has been unified and generalized by the authors in a more general setting combining analytic tools and model theory to ${\rm FO}$-limits (and $X$-limits) and to the notion of modeling. The existence of modeling limits was established for sequences in a bounded degree class and, in addition, to the case of classes of trees with bounded height and of graphs with bounded tree depth. The natural obstacle for the existence of modeling limit for a monotone class of graphs is the nowhere dense property and it has been conjectured that this is a sufficient condition. Extending earlier results here we derive several general results which present a realistic approach to this conjecture. As an example we then prove that the class of all finite trees admits modeling limits.