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 $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_\ell\mathcal{F}|\geq\binom{t}{\ell}$.

The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, 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 $\delta(\mathcal{F}) \geq \binom{t}{k-1}$.

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 $k > \ell \geq 2$.

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 $K^{k}_{t+1}$ whenever $|X| > \frac{1}{4}(t+1)^2\binom{t-1}{\ell-2} + 3t+1$.

In the case when $k=3$ and $\ell=2$, this gives the threshold $\frac14(t+1)^2+3t+1$, which is smaller than $\frac14(t+1)^2(t+2)$ for every $t\ge3$; for $t=2$, the two thresholds give the same integer condition on $|X|$.