  A triple system is cancellative if no three of its distinct edges satisfy $A
\cup B=A \cup C$. It is tripartite if it has a vertex partition into three
parts such that every edge has exactly one point in each part. It is easy to
see that every tripartite triple system is cancellative. We prove that almost
all cancellative triple systems with vertex set [n] are tripartite. This
sharpens a theorem of Nagle and Rodl on the number of cancellative triple
systems. It also extends recent work of Person and Schacht who proved a similar
result for triple systems without the Fano configuration. Our proof uses the
hypergraph regularity lemma of Frankl and Rodl, and a stability theorem for
cancellative triple systems due to Keevash and the second author.
