  A transformation monoid on a set Omega is called synchronizing if it contains
an element of rank 1 (that is, mapping the whole of Omega to a single point).
In this paper, I tackle the question: given n and k, what is the probability
that the submonoid of the full transformation monoid T_n generated by k random
transformations is synchronizing? This question is analogous to Dixon's Theorem
that two random permutations generate the symmetric or alternating group with
high probability.
  Following the technique of Dixon's theorem, we need to analyse the maximal
non-synchronizing submonoids of T_n. I develop a very close connection between
transformation monoids and graphs, from which we obtain a description of
non-synchronizing monoids as endomorphism monoids of graphs satisfying some
very strong conditions. However, counting such graphs, and dealing with the
intersections of their endomorphism monoids, seems difficult.
