--- title: Extremal Graph Theory description: Extremal graph theory studies how global properties of a graph, such as its size or density, force the existence of specific local substructures. Learners will understand Turán's theorem, Ramsey numbers, and how to determine maximum or minimum graph parameters. category: mathematics subcategory: graph-theory difficulty: beginner, intermediate, advanced url: /subject/extremal-graph-theory --- # Extremal Graph Theory Extremal graph theory studies how global properties of a graph, such as its size or density, force the existence of specific local substructures. Learners will understand Turán's theorem, Ramsey numbers, and how to determine maximum or minimum graph parameters. ## Available Resources 2 Books • 3 Courses • 3 Websites • 1 Papers ## Courses ### 1. Graph Theory and Additive Combinatorics (MIT 18.225) **Author:** Yufei Zhao Classical and modern results linking graph theory and additive combinatorics: Turán-type extremal problems, Szemerédi's regularity lemma, pseudorandom graphs, graph limits, Roth's theorem and Freiman's theorem, plus open problems. 26 lecture videos, lecture notes, Zhao's open textbook and problem sets. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-225-graph-theory-and-additive-combinatorics-fall-2023/ **Tags:** extremal-graph-theory, additive-combinatorics, regularity-lemma, pseudorandom-graphs, graph-limits ### 2. Graph Theory and Additive Combinatorics Explore graph theory and additive combinatorics! This course covers key concepts and applications in graphs & digraphs. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX **Tags:** graph-theory, additive-combinatorics, extremal-combinatorics, szemeredi-regularity, lecture-series ### 3. Extremal Graph Theory — Cambridge Part III Course Notes **Author:** David Conlon Fifteen lecture notes from David Conlon's Cambridge Part III course, running from Mantel's and Turán's theorems through Erdős-Stone-Simonovits, the regularity lemma, dependent random choice, Bondy-Simonovits and hypergraph Turán problems. Two example sheets with full solutions. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.its.caltech.edu/~dconlon/Extremal-course.html **Tags:** extremal-graph-theory, regularity-lemma, turan-problems, combinatorics, lecture-notes ## Websites ### 1. Diestel Graph Theory Official site for Reinhard Diestel's Springer graduate text, where the main text is readable free online and paid eBook editions add the full apparatus. It covers matching, connectivity, planarity, colouring, flows, extremal theory, and minors. **Difficulty:** Beginner | **Price:** Free **Link:** https://diestel-graph-theory.com/ **Tags:** graph-theory, textbook, discrete-mathematics, combinatorics, free-online ### 2. Extremal Graph Theory and Ramsey Theory (PCMI 2025 Undergraduate Summer School) **Author:** Yuval Wigderson (notes by Sanjana Das) 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.ias.edu/sites/default/files/uss.pdf **Tags:** extremal-graph-theory, ramsey-theory, combinatorics, lecture-notes ### 3. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Books ### 1. Extremal Graph and Hypergraph Theory: With Ramsey Theory **Author:** Dhruv Mubayi, Jacques A. Verstraëte 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1009585843?tag=edmonddante07-20 **Tags:** extremal-graph-theory, hypergraphs, ramsey-theory, combinatorics, turan-problems ### 2. Extremal Graph Theory **Author:** Béla Bollobás Dover reprint of Bollobás's 1978 monograph, the first systematic treatment of the field: extremal problems on connectivity, matchings, cycles, complete subgraphs, Ramsey theory, and random graphs. Assumes comfort with graph-theoretic proof; still cited as the standard reference. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0486435962?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, graph-theory ## Papers ### 1. The History of Degenerate (Bipartite) Extremal Graph Problems **Author:** Zoltán Füredi, Miklós Simonovits Every textbook treatment stops where Erdős-Stone stops being informative; this is the standard reference for what happens after that, and it is the survey working researchers cite. Ninety-seven-page survey of extremal graph problems where the forbidden graph is bipartite, the degenerate case Erdős-Stone leaves open. Covers Kővári-Sós-Turán, norm graphs, finite-geometry constructions, and the main conjectures, with historical context. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://arxiv.org/abs/1306.5167 **Tags:** extremal-graph-theory, degenerate-extremal-problems, bipartite-graphs, combinatorics, survey --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/extremal-graph-theory