On the distinct maximal-clique sizes in $k$-uniform hypergraphs
Let $g(n,k)$ be the maximum number of distinct sizes of maximal cliques in an $n$-vertex $k$-uniform hypergraph, and let $f(n,k)=n-g(n,k)$. We determine the asymptotic order of $f(n,k)$ for every fixed integer $k\ge 3$. Define $L_2(x)=\max\{2,\log_2(\max\{1,x\})\}$, and, for $j\ge 3$, let $L_j(x)$ be the least number of iterations of $L_{j-1}$ needed to reach a value at most $16$. We prove that $$ f(n,k)=\Theta_k(L_k(n)).$$ In particular, $f(n,3)=\Theta(\log^{*}n)$, where $\log^{*}n$ denotes the iterated logarithm. We also determine the asymptotic behaviour of the layered-tree threshold $c(n,k)$ arising from Gao's insertion-tree method: $$ c(n,k)=\log_2 L_k(n)+O_k(1). $$ Consequently, $f(n,k)=\Theta_k\!\left(2^{c(n,k)}\right)$. Our result gives a negative answer to Gao's question in the case $k=3$.