A note on explicit constructions of designs

TL;DR

This paper presents explicit constructions of certain hypergraphs with small independence numbers, improving upon probabilistic methods for specific parameter ranges and providing deterministic alternatives.

Contribution

The paper introduces explicit constructions of $(n,r,s)$-systems with small independence numbers for cases where $s \\le r/2$, advancing beyond previous probabilistic existence results.

Findings

Explicit $(n,r,s)$-systems with $s \\le r/2$ and $O(n^{1-\\epsilon})$ independence number.

Constructive methods applicable for specific $(r,s)$ pairs, especially with $s \\le r/2$.

Improved deterministic constructions over prior probabilistic existence proofs.

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, R\"{o}dl and \v{S}i\v{n}ajov\'{a} showed that for all fixed integers $r> s \ge 2$, there exists an $(n,r,s)$-system with independence number $O\left(n^{1-\delta+o(1)}\right)$ for some optimal constant $\delta >0$ 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 $O\left(n^{1-\epsilon}\right)$, where $\epsilon > 0$ 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