Can a $k$-planar graph have quadratically many maximal cliques? Can a $k$-planar graph have quadratically many maximal cliques?
I know that any planar or $1$-planar graph $G=(V,E)$ has only $O(|V|)$ maximal cliques, but is there anything known about this question for $k>1$?
 A: As far as I know, this study has only just begun.
The following statement came from Literature [1].

For $k ≥ 3$, not even the maximum number of edges is known for
$n$-vertex $k$-planar graphs, so counting larger cliques is likely to
be an extremely difficult problem.

[1] Gollin J P, Hendrey K, Methuku A, et al. Counting cliques in 1-planar graphs[J]. European Journal of Combinatorics, 2023, 109: 103654.
In fact, Bekos et al. [2] proved that a $3$-planar graph with $n$-vertices has at most $5.5n-10.5$ edges. But we do not know if the bound is tight.
[2] Bekos M A, Kaufmann M, Raftopoulou C N. On the density of non-simple 3-planar graphs[C]//International Symposium on Graph Drawing and Network Visualization. Springer, Cham, 2016: 344-356.
Gollin et al. also give this conjecture:
Conjecture. For $n ≥ 7$, the maximum number of triangles in an $n$-vertex 2-planar graph is at most $(17n − 49)/2$.
The bound in the conjecture  is achieved by the 2-planar graphs formed by stitching copies of $K_7$ together. This stitching is possible since $K_7$ has a $2$-drawing with two facial triangles.

An $n$-vertex 1-planar graph with $4n − 8$ edges is called an optimal 1-planar graph. An open problem for optimal 1-planar graphs that I love is as follows:
Problem. Let $n$ and $t$ be positive integers with $t ∈ \{3, 4, 5\}$ and $n ≥ 10$. What is the maximum number of subgraphs isomorphic to $K_t$
in an $n$-vertex 1-planar graph with $4n − 8$ edges?
To address this problem, I recently worked on the properties of optimal 1-plane graphs, although some are not directly relevant. (Some other progress is being organized)

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*L.C. Zhang, Y.Q. Huang, The reducibility of optimal 1-planar graphs with respect to the lexicographic product [J]. arXiv preprint arXiv:2211.14733, 2022.

