Erdős Problem #775 — Clique Sizes in Three-Uniform Hypergraphs

Does there exist a natural number $C$ such that, for all sufficiently large natural numbers $n$, there is a $3$-uniform hypergraph $H$ on the vertex set $\{0,\ldots,n-1\}$ having at least $n-C$ different sizes of cliques, where cliques are maximal complete subgraphs? Equivalently,

$$n-C\leq\left|\{\text{clique sizes of }H\}\right|.$$

The assertion is false.