Asymptotically sharp bounds for cancellative and union-free hypergraphs

TL;DR

This paper determines the precise asymptotic maximum sizes of certain hypergraphs with cancellative and union-free properties, extending classical results and employing advanced combinatorial frameworks and counting techniques.

Contribution

It establishes exact asymptotic formulas for the maximum edge counts of t-cancellative and t-union-free hypergraphs, refining previous bounds and connecting to longstanding combinatorial conjectures.

Findings

Asymptotic formulas for $C_{2(t-1)}(n,tk)$ and $U_{t+1}(n,tk)$ derived.

Unified framework used to establish lower bounds via hypergraph packings.

Upper bounds obtained through a novel counting argument linked to the Erd ext{"o}s Matching Conjecture.

Abstract

An $r$-graph is called $t$-cancellative if for arbitrary $t+2$ distinct edges $A_1,\ldots,A_t,B,C$, it holds that $(\cup_{i=1}^t A_i)\cup B\neq (\cup_{i=1}^t A_i)\cup C$; it is called $t$-union-free if for arbitrary two distinct subsets $\mathcal{A},\mathcal{B}$, each consisting of at most $t$ edges, it holds that $\cup_{A\in\mathcal{A}} A\neq \cup_{B\in\mathcal{B}} B$. Let $C_t(n,r)$ and $U_t(n,r)$ denote the maximum number of edges that can be contained in an $n$-vertex $t$-cancellative and $t$-union-free $r$-graph, respectively. The study of $C_t(n,r)$ and $U_t(n,r)$ has a long history, dating back to the classic works of Erd\H{o}s and Katona, and Erd\H{o}s and Moser in the 1970s. In 2020, Shangguan and Tamo showed that $C_{2(t-1)}(n,tk)=\Theta(n^k)$ and $U_{t+1}(n,tk)=\Theta(n^k)$ for all $t\ge 2$ and $k\ge 2$. In this paper, we determine the asymptotics of these two functions up to a lower order term, by showing that for all $t\ge 2$ and $k\ge 2$, \begin{align*} \text{$\lim_{n\rightarrow\infty}\frac{C_{2(t-1)}(n,tk)}{n^k}=\lim_{n\rightarrow\infty}\frac{U_{t+1}(n,tk)}{n^k}=\frac{1}{k!}\cdot \frac{1}{\binom{tk-1}{k-1}}$.} \end{align*} Previously, it was only known by a result of F\"uredi in 2012 that $\lim_{n\rightarrow\infty}\frac{C_{2}(n,4)}{n^2}=\frac{1}{6}$. To prove the lower bounds of the limits, we utilize a powerful framework developed recently by Delcourt and Postle, and independently by Glock, Joos, Kim, K\"uhn, and Lichev, which shows the existence of near-optimal hypergraph packings avoiding certain small configurations, and to prove the upper bounds, we apply a novel counting argument that connects $C_{2(t-1)}(n,tk)$ to a classic result of Kleitman and Frankl on a special case of the famous Erd\H{o}s Matching Conjecture.