Universality for and in Induced-Hereditary Graph Properties
Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 1, p. 33.

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The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2א0 ) properties in the lattice K ≤ of induced-hereditary properties of which only at most 2א0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.
Classification : 05C30, 05C63
Mots-clés : countable graph, universal graph, induced-hereditary property 
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Izak Broere; Johannes Heidema. Universality for and in Induced-Hereditary Graph Properties. Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 1, p. 33. https://geodesic-test.mathdoc.fr/item/DMGT_2013__33_1_267920/