A Quadratic Vertex Threshold for Isolated Cliques in the Minimum Degree Kruskal-Katona Problem for 3-Uniform Hypergraphs

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal-Katona theorem states that if $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge \left(\genfrac{}{}{0px}{}{t}{k}\right)$, then $\mathrm{\mid }{\mathrm{\partial }}_{k-1}\mathcal{F}\mathrm{\mid }\ge \left(\genfrac{}{}{0px}{}{t}{k-1}\right)$. The minimum degree version of this problem asks: if $\delta (\mathcal{F})\ge \left(\genfrac{}{}{0px}{}{t}{k-1}\right)$, how small can $\mathrm{\mid }{\mathrm{\partial }}_{k-1}\mathcal{F}\mathrm{\mid }$ be? In this article, for the case $k=3$, we prove that, for every sufficiently large integer $t$, every extremal hypergraph for this problem contains an isolated copy of $K_{t+1}^3$ whenever $\mathrm{\mid }X\mathrm{\mid }\ge c{t}^2+o(t^2)$, with the constant $c=1+\sqrt{928\mathrm{/}33}$. Our proof uses a graph transformation that regularizes the neighborhood structure of extremal graphs, reducing the problem to a counting argument on the neighbors of a disjoint clique family. This gives a quadratic-order threshold for the every-extremal version of the problem, compared with the cubic-order threshold of Füredi and Zhao [SIAM J.\ Discrete Math.\ 36(4), 2022].