We'll discuss a few results concerning triangles in the random graph $G = G(n,p)$, in particular answering (up to a constant factor in the first case):When (ie for what $p= p(n))$ is it true that the maximum sizes of triangle-free and bipartite subgraphs of $G$ coincide?
When is it true that the triangles of $G$ span its cycle space?
The first question dates to Babai, Simonovits and Spencer in 1990, while the second is an instance of a more recent question of M. Kahle.
Joint work with Bobby DeMarco and Arran Hamm.