Year: 2026
Type: preprint
Abstract: We study Turán-type extremal problems for cancellative and locally thin uniform hypergraphs. An $r$-uniform hypergraph is $t$-cancellative if $(\cup_{i=1}^t A_i)\cup B\ne (\cup_{i=1}^t A_i)\cup C$ whenever $A_1,\ldots,A_t,B,C$ are distinct edges. Let $C_t(n,r)$ denote the maximum number of edges in such a hypergraph on $n$ vertices. For all fixed integers $t,k\ge2$, we prove that $C_{2(t-1)}(n,tk)=(1+o(1))\frac{\binom{n}{k}}{\binom{tk-1}{k-1}}$ as $n\to\infty$. In the case $t=2$, this shows that Füredi's 2012 upper bound for $C_2(n,2k)$ is asymptotically sharp. The lower bound uses locally sparse induced packings, while the upper bound follows from double counting and a matching argument. More generally, for integers $s\ge t\ge1$, an $r$-uniform hypergraph is locally $(s,t)$-thin if among any $s$ distinct edges, at least $t$ contain a vertex that lies in none of the other $s-1$ edges. This notion includes cancellative hypergraphs as special cases. We establish general upper and lower bounds for the corresponding extremal numbers and determine their polynomial order of growth under suitable divisibility assumptions.
Source: arXiv (Cornell University)
Authors: Liu, Miao, Shangguan, Chong, Zhang, Chenyang
Institution: Shandong University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4601211488, G7911984978
OpenAlex ID: W7212024338