Lower Bounds for Cover-Free Families

TL;DR

This paper establishes new asymptotic lower bounds on the size of cover-free families, which are combinatorial structures with applications in areas like group testing and coding theory, especially when parameters are within certain bounds.

Contribution

It provides novel asymptotic lower bounds for the minimum size of cover-free families under specific parameter constraints, advancing theoretical understanding.

Findings

Derived new asymptotic lower bounds for (w,r)-cover-free families

Applicable when w ≤ r = |F|^ε for ε ≥ 1/2

Enhances understanding of combinatorial limits in cover-free structures

Abstract

Let ${\cal F}$ be a set of blocks of a $t$-set $X$. $(X,{\cal F})$ is called $(w,r)$-cover-free family ($(w,r)-$CFF) provided that, the intersection of any $w$ blocks in ${\cal F}$ is not contained in the union of any other $r$ blocks in ${\cal F}$. We give new asymptotic lower bounds for the number of minimum points $t$ in a $(w,r)$-CFF when $w\le r=|{\cal F}|^\epsilon$ for some constant $\epsilon\ge 1/2$.