Size of Union

TL;DR

This paper establishes bounds for the size of the union of n sets with empty intersections of any k sets, demonstrating realizability of these bounds and their efficiency in measure and counting versions.

Contribution

It provides new bounds for the union size under specific intersection constraints and shows these bounds are achievable, including in the measure and counting contexts.

Findings

Bounds for union size are established.

Any value between bounds can be realized.

Realizations are most efficient in measure, less so in counting.

Abstract

We find the bounds of the size of the union of n sets satisfying the condition that the intersection of any k sets is empty. We show that any number between the upper and lower bounds can be realised, and in the measure version, the realisation can be the most efficient. The realisation also holds for the counting version, but not always in the most efficient way. The elementary result should be known but surprisingly, we cannot find any reference.