Shadows of Uniform Hypergraphs under a Minimum Degree Condition

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal–Katona theorem implies that if $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge \left(\genfrac{}{}{0px}{}{t}{k}\right)$, then $\mathrm{\mid }{\mathrm{\partial }}_{\mathrm{\ell }}\mathcal{F}\mathrm{\mid }\ge \left(\genfrac{}{}{0px}{}{t}{\mathrm{\ell }}\right)$. The minimum degree version of this problem asks: if $\delta (\mathcal{F})\ge \left(\genfrac{}{}{0px}{}{t}{k-1}\right)$, how small can $\mathrm{\mid }{\mathrm{\partial }}_{\mathrm{\ell }}\mathcal{F}\mathrm{\mid }$ be? We call a hypergraph $\mathit{\text{extremal}}$ if it achieves the minimum value of $\mathrm{\mid }{\mathrm{\partial }}_{\mathrm{\ell }}\mathcal{F}\mathrm{\mid }$ subject to the degree condition $\delta (\mathcal{F})\ge \left(\genfrac{}{}{0px}{}{t}{k-1}\right)$. Füredi and Zhao [SIAM J. Discrete Math. 36(4), 2022] proved that for $k=3$, $\mathrm{\ell }=2$ and $t\ge 2$, every extremal hypergraph contains an isolated copy of $K_{t+1}^3$ when $\mathrm{\mid }X\mathrm{\mid }>\frac{1}{4}(t+1{)}^2(t+2)$. In this article, we study the general case $k>\mathrm{\ell }\ge 2$. By developing a hypergraph transformation that combines shifting operations with antilexicographic compression, we prove that, for every integer $t\ge k-1$, there exists an extremal hypergraph containing an isolated copy of $K_{t+1}^k$ whenever $\mathrm{\mid }X\mathrm{\mid }>\frac{1}{4}(t+1{)}^2\left(\genfrac{}{}{0px}{}{t-1}{\mathrm{\ell }-2}\right)+3t+1$. In the case when $k=3$ and $\mathrm{\ell }=2$, this gives the threshold $\frac{1}{4}(t+1{)}^2+3t+1$, which is smaller than $\frac{1}{4}(t+1{)}^2(t+2)$ for every $t\ge 3$; for $t=2$, the two thresholds give the same integer condition on $\mathrm{\mid }X\mathrm{\mid }$.