This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Extremal graph theory | |
|---|---|
| Name | Extremal graph theory |
| Field | Graph theory |
| Notable figures | Paul Erdős; Pál Turán; Tibor Sós; Alfred Rényi; Vera T. Sós; Béla Bollobás; Paul Turán |
| Related areas | Combinatorics; Ramsey theory; Probabilistic method; Spectral graph theory; Additive number theory |
Extremal graph theory is a branch of Graph theory concerned with determining or estimating the maximum or minimum number of edges in a graph that avoids specified subgraphs or properties. It grew from problems posed by Paul Erdős and classical work of Pál Turán and developed through contributions by researchers associated with institutions such as the Hungarian Academy of Sciences and the University of Cambridge. The subject connects to problems studied by figures like Alfred Rényi, Tibor Sós, and Béla Bollobás and interacts with several themes in modern Combinatorics.
Origins trace to extremal questions studied by Pál Turán in the 1940s and to the prolific problems posed by Paul Erdős in the mid-20th century, often through collaborations with colleagues at the Institute of Mathematics of the Hungarian Academy of Sciences and the Mathematical Institute, Oxford. Motivations included classical results in Number theory and constraints arising in constructions by Alfréd Rényi and studies directed by mentors like Frigyes Riesz and contemporaries such as Vera T. Sós. Later motivation came from problems discussed at workshops hosted by institutions like the Institute for Advanced Study and conferences honoring figures like Paul Turán.
Central questions ask: given n vertices, what is the maximum number of edges in a graph avoiding a particular subgraph H? This led to core concepts developed by researchers at places such as Trinity College, Cambridge and the Courant Institute of Mathematical Sciences: extremal number ex(n,H), Turán graphs, Zarankiewicz problems, and stability theorems. Other notions—graph homomorphisms, graph minors studied at the University of Oxford, and spectral parameters explored at the Princeton University—play foundational roles. Many formulations were influenced by problems circulated by Paul Erdős and formalized in monographs by Béla Bollobás and texts from scholars associated with Cambridge University Press.
Turán’s theorem (1927) provides exact extremal numbers for complete graph avoidance, proved by Pál Turán and influential for later work by Paul Erdős and Vera T. Sós. Generalizations include the Erdős–Stone theorem, developed through collaborations involving scholars at the Mathematical Institute of the Hungarian Academy of Sciences and extended in expository accounts by Béla Bollobás. Zarankiewicz problems, explored by researchers affiliated with Moscow State University and Steklov Institute, address bipartite forbidden subgraphs. Results from researchers at institutions like the University of Cambridge and the Massachusetts Institute of Technology extended Turán-type extremal bounds to hypergraphs and multi-partite constructions.
Ramsey theory originated with Frank P. Ramsey and evolved through landmark contributions by Paul Erdős and George Szekeres, linking unavoidable configurations to extremal counts. Classical Ramsey numbers R(s,t) and generalized hypergraph Ramsey problems were investigated in seminars at the University of Chicago and the Institute for Advanced Study. Probabilistic constructions due to Paul Erdős and refinements by researchers at the California Institute of Technology provide lower bounds, while methods from scholars at the University of Cambridge and the Princeton University yield upper bound techniques and structural theorems.
Key methods include the probabilistic method pioneered by Paul Erdős and colleagues from the Mathematical Institute of the Hungarian Academy of Sciences, analytic techniques using flag algebras developed by researchers associated with Moscow State University and University of Oxford, the regularity lemma by Endre Szemerédi at the Alfréd Rényi Institute, and spectral methods cultivated at the Princeton University and the Massachusetts Institute of Technology. Combinatorial design techniques from specialists at the University of Cambridge and extremal hypergraph methods from teams at the University of Waterloo also play major roles. Techniques often trace to collaborations among figures like Béla Bollobás, Vera T. Sós, and Endre Szemerédi.
Extremal results inform problems in Additive number theory studied at the Institute for Advanced Study and algorithms research at the Massachusetts Institute of Technology, impacting complexity questions investigated at the University of California, Berkeley. Connections to Design theory and coding theory involve work by researchers at the University of Oxford and institutions like Bell Labs; applications to network science link to studies at the Santa Fe Institute and interdisciplinary centers. Cross-disciplinary exchanges occur with scholars in Discrete geometry and theoretical computer science groups at the Carnegie Mellon University.
Major open problems include resolving exact extremal numbers for bipartite graphs posed in problems circulated by Paul Erdős and conjectures related to hypergraph Turán densities discussed in workshops at the Institute of Advanced Study and the Clay Mathematics Institute. The Erdős–Fajtlowicz conjectures, longstanding questions from seminars at the University of Cambridge, and stability conjectures attributed to collaborative networks around Béla Bollobás and Endre Szemerédi remain central. Progress continues through research programs at institutions such as the Mathematical Institute of the Hungarian Academy of Sciences, the Princeton University, and international collaborations supported by bodies like the European Research Council.