  We consider the problem of incrementally maintaining the triangle queries
with arbitrary free variables under single-tuple updates to the input
relations. We introduce an approach called IVM$^\epsilon$ that exhibits a
trade-off between the update time, the space, and the delay for the enumeration
of the query result, such that the update time ranges from the square root to
linear in the database size while the delay ranges from constant to linear
time. IVM$^\epsilon$ achieves Pareto worst-case optimality in the update-delay
space conditioned on the Online Matrix-Vector Multiplication conjecture. It is
strongly Pareto optimal for the triangle queries with zero or three free
variables and weakly Pareto optimal for the triangle queries with one or two
free variables.
