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 $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_{k-1}\mathcal{F}|\geq\binom{t}{k-1}$.

The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, how small can $|\partial_{k-1}\mathcal{F}|$ 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 $|X| \geq ct^2 + o(t^2)$, with the constant $c = 1 + \sqrt{928/33}$.

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].