Broere, IzakHeidema, Johannes2014-05-292014-05-292013Broere, I & Heidema, J 2013, 'Universality for and in induced-hereditary graph properties', Discussiones Mathematicae Graph Theory, vol. 33, no. 1, pp. 34-47.1234-3099 (print)2083-5892 (online)10.7151/dmgt.1671http://hdl.handle.net/2263/39916The 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 20 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(20 ) properties in the lattice K< of induced-hereditary properties of which only at most 20 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.enFaculty of Mathematics, Computer Science and Econometrics, University of Zielona GóraCountable graphUniversal graphInduced-hereditary propertyArticle