STEMMA International Research Journal

Ramsey theory, originating with Ramsey’s 1929 theorem on partitions, has matured into a robust and wide-ranging field of combinatorics. A central and fundamental branch of Ramsey theory studies Ramsey properties for undirected graphs (or 2-colorings of edges), leading naturally to the notion of graph Ramsey numbers. The communication of information among multiple parties can be modeled via multicolor Ramsey properties for many different structures. Although the theory developed considerably during the 20th century, the emergence of fresh perspective in the late 1990s have spurred a resurgence of activity in graph Ramsey theory. Contemporary research remains energized and diverse. New insights, techniques, perspectives, and questions come from applications to dynamical systems, geometric combinatorics, network science, probabilistic methods, theoretical computer science, topology, and beyond. Several conjectures and open problems lie at the heart of current study. Extremal graph Ramsey theory investigates, for a fixed (finite or infinite) graph H, the asymptotic or exact value of n for which every n-vertex graph without an H-subgraph has at least m edges; an H-free graph having edge set E among all n-vertex graphs is called an Hextremal graph. More generally, applications of probabilistic constructions lead to lower bounds on Ramsey numbers involving a prescribed count of edges or vertices. Sparse graph Ramsey theory concerns the edge-density function of the smallest colorsaturation Ramsey graph. Extensive collections of results in sparse structural graph theory, expander graph analysis, and inverse theorems for the Gowersnorm have awakened interest and driven exploration of Ramsey questions within the sparse framework. Multicolor Ramsey theory for graphs investigates, for fixed finite graphs H and C, the smallest integer such that every C-coloring of the edges of the complete graph on n vertices contains a monochromatic copy of H when Results extend to hypergraphs and broader structures. Graphs with additional structure, such as planar or geometric graphs, attract study. Color-restricted Ramsey properties examine forms of monochromatic graphs restricted to particular subfamilies. Connections with combinatorial geometry either arise directly, or emerge through the interplay of hypergraphs and geometric objects.