| Mathematicians use so-called Ramsey numbers to measure how big graphs must get before they inevitably contain such a monochromatic structure, or clique. The Ramsey number R(3) is 6, because a graph must have at least six vertices to guarantee the presence of a red or blue clique of size 3. But Ramsey numbers are notoriously difficult to prove. Mathematicians know that R(4) is 18, but they have yet to compute the exact value of R(5) and beyond. Efforts to solve Ramsey-type problems have led mathematicians to develop some of their most important techniques, like the probabilistic method. Ramsey theory has also been applied to the study of communications networks, information transmission and more. What's New and Noteworthy In the century since Ramsey inadvertently founded Ramsey theory, it's been a particularly active area of research, with several major breakthroughs in just the past few years. Last year, for instance, four mathematicians proved a new, more accurate upper bound on Ramsey numbers — the first advance of its kind since 1935. "I was floored" on hearing the news, one mathematician said. "I was literally shaking for half an hour to an hour." Just a few months later, mathematicians made progress on estimates of asymmetric Ramsey numbers, which deal with graphs that are guaranteed to have red or blue cliques of different sizes. Mathematicians once again found this progress "completely shocking." Also shocking: Some of the mathematicians making headway on these problems are even younger than Ramsey was when he launched the field. In 2020, Quanta wrote about Ashwin Sah, now a graduate student at the Massachusetts Institute of Technology, who as an undergraduate proved major results in Ramsey theory and related areas. Many of these recent breakthroughs involve the study of graphs that grow infinitely large. But mathematicians are also still trying to make sense of small Ramsey numbers, which remain stubbornly elusive. And they're not just looking for monochromatic cliques in graphs; they also want to analyze the emergence of other structures, like branching, treelike patterns, as well as loops called Hamiltonian cycles. In fact, Ramsey theory isn't just about inevitable patterns found in graphs. Hidden structure emerges in lists of numbers, strings of beads and even card games. In 2019, for example, mathematicians studied collections of sets that can always be arranged to resemble the petals of a sunflower. That same year, Quanta reported on research into sets of numbers that are guaranteed to contain numerical patterns called polynomial progressions. And last year, mathematicians proved a similar result, about sets of integers that must always include three evenly spaced numbers, called arithmetic progressions. In its hunt for patterns, Ramsey theory gets to the core of what mathematics is all about: finding beauty and order in the most unexpected places. | 
