Shadows of Uniform Hypergraphs under a Minimum Degree Condition

Abstract

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 $|\mathcal{F}|\ge \left(\genfrac{}{}{0ex}{}{t}{k}\right)$, then $|{\partial }_{\ell }\mathcal{F}|\ge \left(\genfrac{}{}{0ex}{}{t}{\ell }\right)$. The minimum degree version of this problem asks: if $\hat{I}´(\mathcal{F})\ge \left(\genfrac{}{}{0ex}{}{t}{k-1}\right)$, how small can $|{\partial }_{\ell }\mathcal{F}|$ be? We call a hypergraph \textit{extremal} if it achieves the minimum value of $|{\partial }_{\ell }\mathcal{F}|$ subject to the degree condition $\hat{I}´(\mathcal{F})\ge \left(\genfrac{}{}{0ex}{}{t}{k-1}\right)$. Füredi and Zhao [SIAM J. Discrete Math. 36(4), 2022] proved that for $k=3$, $\ell =2$ and $t\ge 2$, every extremal hypergraph contains an isolated copy of $K_{t+1}^3$ when $|X|>\frac{1}{4}(t+1{)}^2(t+2)$. In this article, we study the general case $k>\ell \ge 2$. 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 $|X|>\frac{1}{4}(t+1{)}^2(\genfrac{}{}{0ex}{}{t-1}{\ell -2})+3t+1$. In the case when $k=3$ and $\ell =2$, 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 $|X|$.

14Pages