Uncountable Graphs and Invariant Measures on the Set of Universal Countable Graphs

We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erd\"os--R\'enyi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measures on the space of such graphs. The main idea of our construction is based on the new notions of {\it measurable universal}, and {\it topologically universal} graphs, which are interesting themselves. The realization of the construction can be regarded as two-step randomization for universal measurable graph : {\it "randomization in vertices"} and {\it "randomization in edges"}. For $K_s$-free, $s\geq 3$ there is only randomization in vertices of the measurable graphs. The completeness of our lists is proved using the important theorem by D. Aldous about $S_{\infty}$-invariant matrices, which we reformulate in appropriate way.