Extremal Graph Theory and Hypergraph Problems
Summary
Extremal graph theory seeks the largest or smallest structure a graph can exhibit under given constraints, typically by forbidding a particular subgraph. The classical Turán problem determines the maximum number of edges in an n-vertex graph avoiding a clique of fixed size, while more intricate questions concern forbidden cycles, bipartite subgraphs and Ramsey-type thresholds. Hypergraph generalisations allow edges of size exceeding two, bringing new complexity in uniform and non-uniform settings. Central techniques include stability arguments, probabilistic methods and algebraic constructions. Recent advances have illuminated supersaturation effects, refined bounds for even cycles and disproved long-standing conjectures on hypergraph Lagrangians, with far-reaching applications in combinatorial design, coding theory and theoretical computer science.
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Extremal Graph Theory and Hypergraph Problems publication trend
The graph below shows the total number of articles in extremal graph theory and hypergraph problems across all publications each year (not limited to Nature Index journals).
Technical terms
Turán function (ex(n,F)): Maximum number of edges in an n-vertex graph not containing F as a subgraph.
Uniform hypergraph: A hypergraph in which every hyperedge contains the same number of vertices.
Chromatic index: The minimum number of colours needed to colour hyperedges so that intersecting edges receive different colours.
Lagrangian of a hypergraph: A density-based function maximised by weighting vertices, used to estimate extremal densities.
Supersaturation: The phenomenon where adding edges beyond an extremal threshold forces many copies of a forbidden subgraph.
References
- The Erdös-Faber-Lovász Conjecture for Gap-Restricted Hypergraphs. Engineering (2024).
- Hypergraph Lagrangians I: The Frankl-Füredi conjecture is false. Advances in Mathematics (2020).
- A note on the Turán function of even cycles. Proceedings of the American Mathematical Society (2012).
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