Multicolor Ramsey numbers using random sphere graphs

The multicolor Ramsey number r(t; ℓ) is the minimum integer n such that every ℓ-coloring of the edges of the complete graph Kn contains a monochromatic clique of size t. Improving the asymptotic bounds of r(t; ℓ) for any fixed ℓ ≥ 3 has been a central challenge in combinatorics for nearly a century. The best known lower bounds have traditionally been established using frameworks based on Erd˝os–R´enyi random graphs. Recent advances by Ma, Shen, and Xie in a related problem of the field introduced a random construction known as the random sphere graph. In this thesis, we show that there exists an exponential improvement on the lower bound for multicolor Ramsey numbers by replacing the Erd˝os–R´enyi framework with random sphere graphs. This improvement is achieved by exploiting the inherent geometric constraints of high-dimensional spheres, which make the distribution of cliques and independent sets fundamentally different from those found in other classical random graphs. Furthermore, we provide basic observations on the random sphere model and discuss the potential integration of our method with other emerging frameworks.