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| Mantel's theorem | |
|---|---|
| Name | Mantel's theorem |
| Field | Graph theory |
| Proven | 1907 |
| By | Willem Mantel |
| Statement | Maximum number of edges in a triangle-free graph on n vertices |
Mantel's theorem is a foundational result in extremal Graph theory established by Dutch mathematician Willem Mantel in 1907. It determines the maximum number of edges in a simple triangle-free graph on a given number of vertices and serves as an early prototype for more general extremal results like Turán's theorem and the Erdős–Stone theorem. The theorem has influenced research involving combinatorics in the works of Paul Erdős, Pál Turán, Paul Turán, Erdős–Ko–Rado, and has connections to problems studied by George Szekeres, Lajos Pósa, Alfréd Rényi, and Paul Erdős's collaborators.
Mantel's theorem states that for any simple undirected graph on n vertices that contains no triangle (no 3-cycle), the maximum number of edges is floor(n^2/4). The extremal bound is attained by the complete balanced bipartite graph on n vertices, a configuration akin to those appearing in Turán's theorem and studied by Pál Turán in the context of forbidden complete subgraphs. Mantel's original formulation influenced later work by Béla Bollobás, Endre Szemerédi, László Lovász, Miklós Simonovits, and Béla Bajnok in extremal combinatorics and graph extremal problems.
Multiple proofs of Mantel's theorem appear across the literature, reflecting methods used by researchers such as Paul Erdős, Pál Turán, László Lovász, and Béla Bollobás. Classic proofs include: - A counting argument using double counting and averaging similar to techniques in papers by Paul Erdős and Rademacher. - A proof by induction on n that mirrors approaches in textbooks by László Lovász and Richard Stanley. - A proof via Zykov symmetrization, an operation later formalized by Alexander Zykov and used by Miklós Simonovits in extremal constructions. - A proof using linear algebra and eigenvalues related to methods developed by Erdős, Alfred Rényi, and Fan Chung. Each approach links to methods prominent in works by Paul Turán, Jerzy Neyman style counting, and techniques referenced in surveys by Béla Bollobás, Endre Szemerédi, and Noga Alon.
The extremal graphs attaining the bound are complete bipartite graphs K_{⌊n/2⌋,⌈n/2⌉}, a family familiar from studies by Pál Turán and examples discussed by Paul Erdős and Alfréd Rényi. These configurations are core examples in the classification of extremal graphs developed by Miklós Simonovits and appear in discussions by Béla Bollobás, László Lovász, Noga Alon, and Fan Chung. The stability versions, which describe graphs close to extremal, were advanced by researchers including Erdős, Simonovits, and Endre Szemerédi and relate to later results like the Stability theorem and stability analyses in works by Tibor Szabó and Dániel Marx.
Mantel's theorem is the base case of Turán's theorem (for r=2) and is generalized by the Erdős–Stone theorem, central to extremal combinatorics literature by Paul Erdős, Arthur Stone, and Miklós Simonovits. Other related results include the Zarankiewicz problem studied by Kazimierz Zarankiewicz and elaborated by Béla Bollobás and Noga Alon, the Kővári–Sós–Turán theorem developed by Tibor Kővári, Vera T. Sós, and Pál Turán, and the Erdős–Gallai theorem on paths and cycles by Paul Erdős and Tibor Gallai. Extensions involve hypergraph analogues studied by Paul Erdős, Vera Sós, András Frankl, and Péter Frankl, spectral generalizations in the work of Fan Chung and Alexandre Zvonkin, and probabilistic versions in random graphs by Béla Bollobás and Erdős–Rényi collaborators.
Mantel's theorem underpins numerous applications across combinatorics and theoretical computer science, influencing results by Paul Erdős, Noga Alon, László Babai, Miklós Simonovits, and Endre Szemerédi. It appears in proofs concerning Ramsey-type problems such as those by Frank P. Ramsey and in structural graph theory studies by Paul Erdős and Endre Szemerédi. Algorithmic implications connect to work by Richard M. Karp, Michael Garey, David S. Johnson, and Jon Kleinberg in optimization and approximation. Mantel's insight also informs extremal set theory explored by Erdős–Ko–Rado contributors László Lovász and Peter Frankl and has ramifications in additive combinatorics in research by Terence Tao, Ben Green, and Imre Z. Ruzsa.