Triangular Ramsey Numbers

TL;DR

This paper introduces triangular Ramsey numbers, explores their bounds, and uses a simple combinatorial game called Mines to analyze and define these numbers, providing new insights into their properties.

Contribution

It defines triangular Ramsey numbers and establishes bounds using a novel combinatorial game and probabilistic methods, expanding Ramsey theory into a new geometric context.

Findings

Triangular Ramsey numbers are introduced and bounded.

The game Mines is used to analyze triangular sets and strategies.

Lower bounds for the numbers are established using probabilistic methods.

Abstract

The purpose of this paper is to introduce the idea of triangular Ramsey numbers and provide values as well as upper and lower bounds for them. To do this, the combinatorial game Mines is introduced; after some necessary theorems about triangular sets are proved. This game is easy enough that young children are able to play. The most basic variations of this game are analyzed and theorems about winning strategies and the existence of draws are proved. The game of Mines is then used to define triangular Ramsey numbers. Lower bounds are found for these triangular Ramsey numbers using the probabilistic method and the theorems about triangular sets.