Jan GREBIK: A rigid Urysohn-like metric space

ABSTRACT: We construct a graph of the smallest uncountable cardinality which has the same extension
property as the Random graph, yet its group of automorphisms is trvial. We also present how
similar, although technically more  complicated idea may lead to a construction of a complete
metric space of density aleph1, having the extension property like the Urysohn space, yet
again its group of isometries is trivial. Finally we discuss possible universal properties of this
structures and possible generalization to the Gurarij space.