On the boundedness of degenerate hypergraphs

TL;DR

This paper studies the boundedness property of degenerate hypergraphs, showing many well-known classes are bounded, which is important for extending classical combinatorial theorems, and introduces new results on Zarankiewicz-type problems for 3-graphs.

Contribution

It proves the boundedness of various degenerate hypergraphs, including even cycles and bipartite graphs, and introduces a new Zarankiewicz-type problem for 3-graphs.

Findings

Many degenerate hypergraphs are bounded.

Boundedness is key for extending classical theorems.

Introduces a new Zarankiewicz-type problem for 3-graphs.

Abstract

We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex $F$-free $r$-graph with a vertex of degree at least $\alpha \binom{n-1}{r-1}$ has fewer than $(1-\beta) \cdot \mathrm{ex}(n,F)$ edges. The boundedness property is crucial for recent works~\cite{HHLLYZ23a,DHLY24} that aim to extend the classical Hajnal--Szemer\'{e}di Theorem and the anti-Ramsey theorems of Erd\H{o}s--Simonovits--S\'{o}s. We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for $3$-graphs, strengthening a theorem by Kostochka--Mubayi--Verstra\"{e}te~\cite{KMV15}.