This paper provides an overview of Euclidean Ramsey theory and the challenges associated with determining the chromatic number of space. We concentrated on lowering the upper bounds of chromatic numbers for Euclidean spaces of lower dimensions. We introduced and utilized concepts such as lattices, sublattices, and Voronoi cells, which helped us achieve colorings omitting distance 1 of particular spaces using fewer colors than previously documented. Specifically, we established that \(\mathcal{X}(\mathbb{E}^3) \leq 15\) by applying modular coloring techniques. Moreover, we established the relationship between the lattice’s covering and packing ratios and the chromatic number of the associated \(n\)-dimensional space. This approach allowed us to determine that for \(n \leq 17\), the chromatic number satisfies \(\mathcal{X}(\mathbb{E}^n) \leq 3^n\), with further findings extending this to \(24 \leq n \leq 33\) and \(n = 48\). Additionally, we expanded the definitions of lattices and sublattices to complex spaces. Through the application of complex Gaussian lattices, we decreased the upper bound from \(3^n\) to \(\sqrt{8}^n\) for several even-dimensional Euclidean spaces and further reduced it to \(\sqrt{7}^n\) using Eisenstein lattices. The paper outlines definitions for all utilized lattices and highlights the principal properties that were leveraged. We conclude by discussing unresolved questions and potential areas for further research.