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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Recent work on hypergraph colourings has validated the Erdős–Faber–Lovász conjecture within a broad class of gap-restricted hypergraphs, establishing that the chromatic index equals the vertex count for this family and extending earlier partial results on intersecting set systems.
A landmark study in Lagrangian theory has refuted the Frankl–Füredi conjecture by constructing an infinite family of r-uniform hypergraphs whose Lagrangian values exceed those of the colex initial segment. The authors further identify parameter ranges in which the original conjecture still holds.
In graph theory, a refined upper bound for the Turán function of even cycles has been achieved, showing that ex(n,C2k) scales as (k−1)n1+1/k plus lower-order terms. This result narrows the gap to the conjectured asymptotic and leverages a combination of combinatorial and analytic methods.
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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