a R b <--> a and b have visited the same cities.

is R transitive i.e. (aRb and bRc -->aRc) ?

1) No.
suppose

a has visited Rome, paris, london
b has visited rome, paris, New York
c has visited paris, New York, Honolulu

then aRb since they have visited the same cities (Rome, Paris)
bRc (paris, New York)

but a is not related to c since they have only one city in common not "cities" as required.

2) Yes.

"have visited the same cities" means that the set of cities visited by a and b is the same. obviously trnasitive.
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Basically the question reduces to wether there is an invisible "all" sitting in the phrase "have visitied (all) the same cities."

I think in math the convention is that there is never an implicit all, and in English in this case, there is an implicit "all."

what do you (all) think?

If you wanted to phrase this so that I did not think you meant "all", you would want to say "a R b if a and b have visited cities in common" or "visited some cities together".

You do raise a good point, however, in making sure that one is able to correctly distinguish universal from existential quantifiers, particularly in plain language.