# The discrete yet ubiquitous theorems of Carath\'eodory, Helly, Sperner, Tucker, and Tverberg

## TL;DR

This paper explores five fundamental discrete theorems from combinatorial topology and geometry, highlighting their connections and broad applications across various fields like game theory and optimization.

## Contribution

It provides a unified discussion of five key theorems, emphasizing their interrelations and significance in multiple application domains.

## Key findings

- Identifies connections among Sperner, Tucker, Carath\'eodory, Helly, and Tverberg theorems.
- Highlights their broad impact in fields such as game theory, graph theory, and computational geometry.
- Emphasizes the importance of these theorems in practical applications.

## Abstract

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc.