In the distributed subgraph-freeness problem, we are given a graph $H$, and asked to determine whether the network graph contains $H$ as a subgraph or not. Subgraph-freeness is an extremely local problem: if the network had no bandwidth constraints, we could detect any subgraph $H$ in $|H|$ rounds, by having each node of the network learn its entire $|H|$-neighborhood. However, when bandwidth is limited, the problem becomes harder. Upper and lower bounds in the presence of congestion have been established for several classes of subgraphs, including cycles, trees, and more complicated subgraphs. All bounds shown so far have been linear or sublinear. We show that the subgraph-freeness problem is not, in general, solvable in linear time: for any $k geq 2$, there exists a subgraph $H_k$ such that $H_k$-freeness requires $Omega( n^{2-1/k} / (Bk) )$ rounds to solve. Here $B$ is the bandwidth of each communication link. The lower bound holds even for diameter-3 subgraphs and diameter-3 network graphs. In particular, taking $k = Theta(log n)$, we obtain a lower bound of $Omega(n^2 / (B log n))$.
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