Sharp bounds for uniform union-free hypergraphs

TL;DR

This paper determines the asymptotic maximum size of $t$-union-free $r$-uniform hypergraphs for most parameters, advancing the understanding of their extremal properties and introducing near-optimal hypergraph packings.

Contribution

It provides the asymptotic behavior of $U_t(n,r)$ for almost all $t extgreater=3$ and $r extgreater=3$, a significant extension beyond previous known cases.

Findings

Asymptotic formula for $U_t(n,r)$ for most $t,r$

Existence of near-optimal locally sparse hypergraph packings

Progress in understanding extremal union-free hypergraphs

Abstract

An $r$-uniform hypergraph is called $t$-union-free if any two distinct subsets of at most $t$ edges have distinct union. The study of union-free hypergraphs has multiple origins and a long history, dating back to the works of Kautz and Singleton (1964) in coding theory, Bollob\'as and Erd\H{o}s (1976) in combinatorics, and Hwang and S\'os (1987) in group testing. Let $U_t(n,r)$ denote the maximum number of edges in an $n$-vertex $t$-union-free $r$-uniform hypergraph. In this paper, we determine the asymptotic behavior of $U_t(n,r)$, up to a lower order term, for almost all $t\ge 3$ and $r\ge 3$. This significantly advances the understanding of this extremal function, as previously, only the asymptotics of $U_2(n,3)$ and $U_2(n,4)$ were known. As a key ingredient of our proof, we establish the existence of near-optimal locally sparse induced hypergraph packings, which is of independent interest.