$G$ and an integer $k$, the task is to count the number of induced subgraphs on

$k$ vertices that satisfy the graph property $\\Phi$. Focke and Roth [STOC 2022]

completely characterized the complexity for each $\\Phi$ that is a hereditary

property (that is, closed under vertex deletions): #IndSub($\\Phi$) is

#W[1]-hard except in the degenerate cases when every graph satisfies $\\Phi$ or

only finitely many graphs satisfy $\\Phi$. We complement this result with a

classification for each $\\Phi$ that is edge monotone (that is, closed under

edge deletions): #IndSub($\\Phi$) is #W[1]-hard except in the degenerate case

when there are only finitely many integers $k$ such that $\\Phi$ is nontrivial

on $k$-vertex graphs. Our result generalizes earlier results for specific

properties $\\Phi$ that are related to the connectivity or density of the graph.

Further, we extend the #W[1]-hardness result by a lower bound which shows

that #IndSub($\\Phi$) cannot be solved in time $f(k) \\cdot |V(G)|^{o(\\sqrt{\\log

k/\\log\\log k})}$ for any function $f$, unless the Exponential-Time Hypothesis

(ETH) fails. For many natural properties, we obtain even a tight bound $f(k)

\\cdot |V(G)|^{o(k)}$; for example, this is the case for every property $\\Phi$

that is nontrivial on $k$-vertex graphs for each $k$ greater than some $k_0$.

},\n}\n'