we consider a generalization of the maximum degree in random graphs. given a rooted tree $t$, let $x_v$ denote the number of copies of t rooted at v in the binomial random graph $g_{n,p}$. we ask the question where the maximum $m_n = max \{x_1, ..., x_n\}$ of $x_v$ over all vertices is concentrated. for edge-probabilities $p=p(n)$ tending to zero not too fast, the maximum is asymptotically attained by the vertex of maximum degree. however, for smaller edge probabilities $p=p(n)$, the behavior is more complicated: our large deviation type optimization arguments reveal that the behavior of $m_n$ changes as we vary $p=p(n)$, due to different mechanisms that can make the maximum large.

based on joint work with pedro araújo, simon griffiths and matas Šileikis; see arxiv:2310.11661