Modeling Limits in Hereditary Classes: Reduction and Application to Trees

Jaroslav Nesetril (1) , Patrice Ossona de Mendez (1, 2)

1 IUUK - Computer Science Institute of Charles University [Prague]
2 CAMS - Centre d'Analyse et de Mathématique sociales

Limits of graphs were initiated recently in the two extreme contexts of dense and bounded degree graphs. This led to elegant limiting structures called graphons and graphings. These approach have been unified and generalized by authors in a more general setting using a combination of analytic tools and model theory to 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. These seemingly very special classes is in fact a key step in the development of limits for more general situations. 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 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.

Mots clés: Graph; Relational structure; Graph limits; Structural limits; Stone space; Model theory; First-order logic; Measurable graph

Domaines: Combinatoire [math.CO]

Jaroslav Nesetril, Patrice Ossona de Mendez. Modeling Limits in Hereditary Classes: Reduction and Application to Trees. 2013. ⟨hal-00912153⟩