An Exponential Improvement for Diagonal Ramsey
by Marcelo Campos, Simon Griffiths, Robert Morris, Julian Sahasrabudhe · arXiv
Research paper proving R(k) is at most (4 minus epsilon) to the k, the first exponential improvement on the Erdős-Szekeres upper bound of 1935. Introduces the book algorithm for locating cliques in two-colored complete graphs. The endpoint of the ladder: a learner who has worked through the course and surveys should read the paper that changed the headline bound, and its introduction is unusually readable about why the Erdős-Szekeres argument resisted improvement for so long.
More resources on Ramsey Theory
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Extremal Graph and Hypergraph Theory: With Ramsey Theory
Cambridge Studies in Advanced Mathematics text covering Turán-type problems for graphs and hypergraphs and Ramsey theory, using probabilistic and algebraic methods, with complete proofs of recent results on sunflowers and off-diagonal and geometric Ramsey numbers. Readers can follow and apply current research techniques.
Extremal Graph Theory and Ramsey Theory (PCMI 2025 Undergraduate Summer School)
The only genuinely accessible on-ramp found that is still fully rigorous: it assumes no graduate background, builds extremal numbers from Mantel forward, and proves Erdős-Stone-Simonovits and Kővári-Sós-Turán in full. Eighty pages of notes from Yuval Wigderson's 2025 PCMI undergraduate summer school, working from extremal numbers and Erdős-Stone-Simonovits through Kővári-Sós-Turán, supersaturation and stability, then continuing into graph and hypergraph Ramsey numbers.
Small Ramsey Numbers (Dynamic Survey DS1)
Every other source defers to this document for actual numeric values. Continuously revised reference compiling all known nontrivial values and bounds for two-color, multicolor, graph and hypergraph Ramsey numbers, with citations to the constructions and computations behind each entry. Latest revision April 2026.
Recent Developments in Graph Ramsey Theory
The standard orientation document for anyone moving from a course to the literature: it states what is known, what the proof techniques are, and which problems are open, with a bibliography that functions as the field's reading list. Fifty-four page survey of graph Ramsey theory covering diagonal and off-diagonal bounds, Ramsey numbers of sparse and degenerate graphs, hypergraph and induced Ramsey problems, and Ramsey-Turán questions, with open problems. Published in Surveys in Combinatorics 2015.
Ramsey Theory (ETH Zürich 401-4054, Spring 2024)
ETH Zürich graduate course page with complete lecture notes, recordings, twelve problem sets and exam solutions, covering classical bounds, off-diagonal and multicolor Ramsey numbers, the regularity method, induced and canonical Ramsey theorems, and Hales-Jewett.