On the distinct maximal-clique sizes in $3$-uniform hypergraphs

Let $g(n,k)$ be the largest possible number of distinct sizes of maximal cliques in a $k$-uniform hypergraph on $n$ vertices, and let $f(n,k)=n-g(n,k)$. In the graph case, Spencer proved in 1971 that $f(n,2)=\Theta(\log n)$. For $3$-uniform hypergraphs, Erd\H{o}s constructed examples showing that $f(n,3)\le \log^* n+O(1)$, where $\log^* n$ is the number of iterated logarithms such that $\log\log \ldots \log n<1$. Recently, Gao (JCT-B, 2026) proved that $f(n,3)\ge\log \log^* n-O(1)$, thereby answering a question posed by Erd\H{o}s. In the same paper, Gao defined the associated layered-tree threshold $c(n,3)$ and asked whether $f(n,k)=\Theta(c(n,3))$. In this paper, we determine the correct order $$f(n,3)=\Theta(\log^* n).$$ Since $c(n,3)=\log\log^* n+O(1)$, our result gives a negative answer to Gao's question in the case $k=3$. We conclude by proposing the following conjecture: $f(n,k)=\Theta_k\!\left(2^{c(n,k)}\right)$ for every fixed $k\ge3$.