Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)?

The answer is no, as proved by Gao [Ga25]: more generally, for any $k\geq 3$, every $k$-uniform hypergraph on $n$ vertices contains at most $n-f_k(n)$ different sizes of cliques, where $f_k(n)\to \infty$ as $n\to \infty$.