Executive Summary : | This project aims to explore fundamental problems in the Zero-Sum Ramsey Theory branch of Ramsey Theory. The main focus is on finding balanced subgraphs, which have equal numbers of edges of each color. The central question is when a Red-Blue graph G can contain a balanced subgraph on k edges satisfying a particular property and how efficiently it can be found. The chosen properties include spanning tree, spanning path, completeness, connectedness, r-connectedness, acyclicity, bipartiteness, and r-regularity. The project is theoretical in nature, and the findings will be codified as theorems with rigorous proofs. Additionally, algorithms will be designed, providing rigorous proof of correctness and guarantees on their running times. The goal is to contribute significantly to understanding an interesting class of problems in Zero-sum Ramsey Theory and to develop a toolkit of an ensemble of algebraic, combinatorial, and algorithmic techniques. The project aims to contribute to the understanding of this area and contribute to the field of Ramsey Theory. |