J. Gao, On cliques in hypergraphs. arXiv:2510.14804 (2025).

Moon, J. W. and Moser, L., On cliques in graphs. Israel J. Math. 3 (1965), no. 1, 23--28, doi:10.1007/BF02760024. The graph case: $g(n)$, the maximum number of different sizes of cliques (maximal complete subgraphs) in a graph on $n$ nodes (p. 23), with Theorem 3 (p. 25), $g(n)\ge n-[\log_2n]-2[\log_2\log_2n]-4$ for $n\ge26$, and Theorem 4 (p. 27), $g(n)\le n-[\log_2n]$ for $n\ge4$; the paper has no hypergraph statement. Library home: Moon and Moser 1965; paged at Moon and Moser 1965, Theorem 3 (p. 25) and Moon and Moser 1965, Theorem 4 (p. 27).

Spencer, J. H., On cliques in graphs. Israel J. Math. 9 (1971), no. 4, 419--421, doi:10.1007/BF02771457. The graph case: with cliques the maximal complete subgraphs and logarithms to the base $2$, "for $N$ sufficiently large ($>33000$ will do) $g(N)\ge N-\log N-4$" (p. 419), the lower bound that meets Moon and Moser's Theorem 4 up to a constant; the paper has no hypergraph statement. Library home: Spencer 1971; paged at Spencer 1971, Main bound (p. 419).