OPEN
Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)
Erdős constructed such a hypergraph with cliques of at least $n-\log_*n$ different sizes. For graphs, Spencer
[Sp71] constructed a graph which contains cliques of at least $n-\log_2n+O(1)$ different sizes, which Moon and Moser
[MoMo65] showed to be best possible.
See also [927].
