  In this paper, we show that there is a close relation between consistency in
a constraint network and set intersection. A proof schema is provided as a
generic way to obtain consistency properties from properties on set
intersection. This approach not only simplifies the understanding of and
unifies many existing consistency results, but also directs the study of
consistency to that of set intersection properties in many situations, as
demonstrated by the results on the convexity and tightness of constraints in
this paper. Specifically, we identify a new class of tree convex constraints
where local consistency ensures global consistency. This generalizes row convex
constraints. Various consistency results are also obtained on constraint
networks where only some, in contrast to all in the existing work,constraints
are tight.
