Explicit geometric construction of triangle-free Ramsey graphs

We describe an explicit geometric construction of a vast family of graphs without m-cliques with bounded independence number generalizing triangle-free Ramsey graphs described by Codenotti, Pudlák and Resta and provide a new combinatorial proof for the upper bound on the independence number of the latter. We focus on triangle-free graphs and describe some families of such graphs with n vertices and independence number O (n^2{3}) which give us a constructive asymptotic lower bound Ω (t^3{2}) for Ramsey numbers R (3, t) which achieves the best-known constructive lower bound. We describe an additional family of graphs that don't match the best-known bound but still have a polynomial independence number with regards to the number of vertices and are based on Euclidean geometry. We also present a linear 12-approximation algorithm for finding the largest independent set that works for a significant subset of our family of graphs.