Erdős–Stone theorem

Erdős–Stone theorem The Erdős–Stone theorem is a fundamental result in extremal Paul Erdős Graph theory that determines the asymptotic maximum number of edges in an n-vertex graph excluding a fixed non-bipartite subgraph H. It links extremal graph theory to chromatic properties, resolving a longstanding class of problems posed by Turán's theorem and extending work of Turán and Mantel. The theorem is often described as the "fundamental theorem of extremal graph theory" and has deep connections to problems studied by Rademacher, Kazimierz Zarankiewicz, Erdős–Rényi model, and results in Ramsey theory.

Statement

For every fixed graph H with chromatic number r = χ(H) ≥ 2 and every ε > 0, for sufficiently large n any n-vertex graph G with more than (1 − 1/(r−1) + ε) * C(n,2) edges necessarily contains H as a subgraph. Equivalently, the extremal number ex(n,H) satisfies ex(n,H) = (1 − 1/(r−1) + o(1)) * C(n,2), where r = χ(H). This reduces the asymptotic determination of ex(n,H) to the chromatic number of H and refines Mantel's theorem (the case r = 2) and Turán's theorem (for complete graphs). The theorem distinguishes the bipartite case (χ(H) = 2), where the bound is o(n^2) only in special instances, from non-bipartite cases governed by the Turán density 1 − 1/(r−1).

History and development

The theorem emerged from collaborations and exchanges among Paul Erdős, Arthur Stone, and contemporaries in the 1940s; it was announced in papers around 1946–1947. It built on earlier work by Pál Turán (Turán's theorem, 1941) and Vojtěch Jarník-era problems studied by Otakar Borůvka and others in Central European circles. Subsequent refinements and rigorous expositions involved contributors such as Béla Bollobás, András Hajnal, Miklós Simonovits, and Vojtěch Rödl, who placed the result within a broader program of extremal combinatorics developed alongside advances by Alfréd Rényi, Erdős–Rényi, and later researchers linked to the Szemerédi regularity lemma and Hypergraph Turán problems. The intellectual lineage connects to prizes and institutions that fostered combinatorics, including work circulated at Institute for Advanced Study, Princeton University, Hungarian Academy of Sciences, and gatherings like the Colloquium of the International Mathematical Union.

Proof sketch and methods

The original proof uses averaging arguments, probabilistic constructions, and combinatorial counting initiated by Paul Erdős and coauthors, combined with extremal constructions inspired by Turán and Mantel. A typical approach partitions the vertex set into r−1 parts to compare G with the Turán complete (r−1)-partite graph and applies deletion and counting lemmas to force an H-copy when edge density exceeds the Turán threshold. Modern expositions often invoke the Szemerédi regularity lemma and the blow-up lemma of Komlós, Sárközy, and Szemerédi to give more structured embeddings; alternate routes use supersaturation results developed by Erdős and Simonovits, along with stability versions proved by Simonovits that show near extremal graphs are close to Turán graphs. Probabilistic methods from Erdős–Rényi model and concentration inequalities by Paul Erdős collaborators such as Joel Spencer also streamline arguments. For bipartite exceptions, methods extend to use Kővári–Sós–Turán theorem techniques and algebraic constructions by Elekes and Bukh.

Applications and consequences

The theorem has numerous applications in extremal graph theory and related fields. It gives asymptotic answers to forbidden-subgraph problems posed in texts by Béla Bollobás and informs bounds in Turán-type problems for hypergraphs as studied by Turán-followers such as Erdős, Frankl, and Rödl. It underpins stability results used in proofs of exact extremal numbers for families of graphs by Simonovits, Nikiforov, and Keevash. Connections reach into Ramsey theory where chromatic thresholds influence Ramsey multiplicities explored by Erdős and Sós, and into additive combinatorics via graph representations used by Elekes and Tao. Algorithmic consequences appear in property testing frameworks developed by Alon and Shapira, while probabilistic combinatorics and random graph thresholds studied by Bollobás and Janson exploit the density dichotomy given by the theorem. The result informs extremal questions in finite geometry and design theory investigated by Erdős associates and organizations like American Mathematical Society meetings.

Related results and generalizations

Generalizations include stability theorems by Miklós Simonovits, sparse random analogues proved by Conlon, Gowers, and Schacht, and hypergraph extensions in extremal set theory by Turán-era researchers such as Frankl and Füredi. The Erdős–Stone–Simonovits theorem refines the original by adding explicit error terms and stability characterizations; further extensions address ordered graphs studied by Pach and Tóth and induced forbidden subgraph problems linked to work by Alon, Fox, and Sudakov. Bipartite cases spur separate lines of research including the Kővári–Sós–Turán theorem, Zarankiewicz problem developments by Füredi and Nikiforov, and constructions from algebraic and number-theoretic methods by Bukh and Blokhuis. Ongoing research at institutions like Massachusetts Institute of Technology, University of Cambridge, and Princeton University advances tight bounds, stability versions, and algorithmic implications.

Category:Theorems in graph theory