Monochromatic Edges in Complete Multipartite Hypergraphs

TL;DR

This paper investigates the minimum number of same-gender student triples across multiple classes in a school, generalizing the problem to various class sizes, group sizes, and attributes.

Contribution

It provides solutions to generalized versions of Muaengwaeng's problem, including multiple classes, larger groups, unequal class sizes, and non-binary attributes.

Findings

Solved for multiple classes and larger group sizes

Extended to classes of different sizes

Included non-binary attribute scenarios

Abstract

Consider the following problem. In a school with three classes containing $n$ students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occurs when the classes are all boy, all girl and half-and-half. In this paper, we solve many generalizations of the problem including when the school has more than three classes, when triples are replaced by groups of larger sizes, when the classes are of different sizes, and when gender is replaced by other non-binary attributes.