  A Rado simplicial complex X is a generalisation of the well-known Rado graph.
X is a countable simplicial complex which contains any countable simplicial
complex as its induced subcomplex. The Rado simplicial complex is highly
symmetric, it is homogeneous: any isomorphism between finite induced
subcomplexes can be extended to an isomorphism of the whole complex. We show
that the Rado complex X is unique up to isomorphism and suggest several
explicit constructions. We also show that a random simplicial complex on
countably many vertices is a Rado complex with probability 1. The geometric
realisation |X| of a Rado complex is contractible and is homeomorphic to an
infinite dimensional simplex. We also prove several other interesting
properties of the Rado complex X, for example we show that removing any finite
set of simplexes of X gives a complex isomorphic to X.
