Non-trivial $r$-wise agreeing families

TL;DR

This paper investigates the maximum size of non-trivial r-wise agreeing families of subsets, generalizing classical combinatorial theorems, with implications for discrete optimization.

Contribution

It determines the largest size of non-trivial r-wise agreeing families, extending the Brace-Daykin theorem to broader cases.

Findings

Established the maximum size of non-trivial r-wise agreeing families.

Generalized the classical Brace-Daykin theorem.

Provides new bounds relevant to discrete optimization.

Abstract

A family of subsets of $[n]$ is $r$-wise agreeing if for any $r$ sets from the family there is an element $x$ that is either contained in all or contained in none of the $r$ sets. The study of such families is motivated by questions in discrete optimization. In this paper, we determine the size of the largest non-trivial $r$-wise agreeing family. This can be seen as a generalization of the classical Brace-Daykin theorem.