Generalising separating families of fixed size

TL;DR

This paper investigates the minimal number of non-adaptive tests needed to identify a fixed-size subset within a larger set, providing asymptotically optimal bounds for test counts as the set size grows.

Contribution

It establishes asymptotically sharp bounds on the minimum number of tests required for fixed subset size and test size in a classic combinatorial search problem.

Findings

Derived asymptotically sharp bounds for test numbers

Analyzed the non-adaptive testing model with fixed subset and test sizes

Provided results as the set size tends to infinity

Abstract

We examine the following version of a classic combinatorial search problem introduced by R\'enyi: Given a finite set $X$ of $n$ elements we want to identify an unknown subset $Y \subset X$ of exactly $d$ elements by testing, by as few as possible subsets $A$ of $X$, whether $A$ contains an element of $Y$ or not. We are primarily concerned with the model where the family of test sets is specified in advance (non-adaptive) and each test set is of size at most a given $k$. Our main results are asymptotically sharp bounds on the minimum number of tests necessary for fixed $d$ and $k$ and for $n$ tending to infinity.