Geodesic
Universality for and in Induced-Hereditary Graph Properties
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 33-47

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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^ℵ_0 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 2^2^ℵ_0 properties in the lattice 𝕂_≤ of induced-hereditary properties of which only at most 2^ℵ_0 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.
Keywords: countable graph, universal graph, induced-hereditary property 
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Broere, Izak; Heidema, Johannes. Universality for and in Induced-Hereditary Graph Properties. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 33-47. http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a3/