Show that there doesn't exist a (countable) universal graph for each of the following classes of graphs (that is, no universal graph for class 1, and unrelatedly, no universal graph for class 2):
All graphs that do not contain the countable complete graph.
All graphs where the degree of each vertex is finite.
So a proof would involve showing that for every graph $G$ in one of these class, there is a graph in that same class $G'$ that is not isomorphic to some induced subgraph of $G$. How would I go about doing this? I know of the Rado random graph, which contains every countable graph as an induced subgraph, what fails in the two classes above (as opposed to all countable graphs) to make it so that there is no universal graph?