We consider the problem of testing whether a graph on $n$ vertices
and average
degree $d$ is triangle-free. The algorithm should accept graphs that
are
triangle-free and reject graphs that are far from being triangle-free
in
the sense that a large fraction of the edges should be removed
in order to obtain a triangle-free graph. The algorithm is allowed
a small probability of error. This problem has been studied quite
extensively
in the past, but the focus was on dense graphs, that is, when
$d = \Theta(n)$.
Our main finding is a lower bound of $\Omega(n^{1/3})$ on the
necessary
number of queries, where this bound holds for *every* $d <
n^{1-o(1)}$.
Since when $d = \Theta(n)$ the number of queries sufficient for
testing does not depend on $n$, we observe an abrupt,
*threshold-like*
behavior of the complexity of testing around $n$.
In the course of our analysis we show that dense random Cayley
graphs behave like quasi-random graphs in the sense that relatively
large
subsets of vertices have the ``correct'' edge density. The result for
subsets of this size cannot be obtained from the known spectral
techniques
that only supply such estimates for much larger subsets.
This is joint work with Noga Alon, Michael Krivelevich and Dana Ron.