Year: 2021
Type: preprint
Abstract: An $(n,r,s)$-system is an $r$-uniform hypergraph on $n$ vertices such that every pair of edges has an intersection of size less than $s$. Using probabilistic arguments, Rödl and Šiňajová showed that for all fixed integers $r> s \ge 2$, there exists an $(n,r,s)$-system with independence number $O\left(n^{1-δ+o(1)}\right)$ for some optimal constant $δ>0$ only related to $r$ and $s$. We show that for certain pairs $(r,s)$ with $s\le r/2$ there exists an explicit construction of an $(n,r,s)$-system with independence number $O\left(n^{1-ε}\right)$, where $ε> 0$ is an absolute constant only related to $r$ and $s$. Previously this was known only for $s>r/2$ by results of Chattopadhyay and Goodman
Source: arXiv (Cornell University)
Authors: Liu, Xizhi, Mubayi, Dhruv
Institution: University of Illinois Chicago
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3170155768