An Improved Threshold for the Minimum Degree Kruskal-Katona Theorem for 3-Uniform Hypergraphs

TL;DR

This paper improves the threshold for the minimum degree version of the Kruskal-Katona theorem in 3-uniform hypergraphs, showing extremal graphs contain a large clique when the vertex set size exceeds a quadratic function of t.

Contribution

It establishes a tighter quadratic threshold for extremal graphs in the minimum degree Kruskal-Katona problem for 3-uniform hypergraphs, refining previous cubic bounds.

Findings

Extremal graphs contain an isolated K_{t+1}^3 when |X| ≥ c t^2 + o(t^2)

The threshold is reduced from O(t^3) to O(t^2)

The proof involves a graph transformation to analyze neighborhood structures

Abstract

Given a set $X$ and a sufficiently large 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 every extremal graph 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 improves a result of F\"{u}redi and Zhao [SIAM J.\ Discrete Math.\ 36(4), 2022], reducing the threshold from $O(t^3)$ to $O(t^2)$.