Extremal and probabilistic combinatorics
How large can a discrete structure be without containing a forbidden pattern? Ramsey theory, Turán problems and set systems, attacked mostly by random constructions, entropy and iterative algorithms.
Extremal combinatorics supplies the quantitative backbone of graph theory, theoretical computer science (pseudorandomness, property testing, communication complexity) and additive combinatorics. Its methods, the probabilistic method and entropy arguments, spread widely.
How large can a structure get before a given pattern is forced to appear?
- 1Ask how large a structure can avoid a pattern
- 2Large enough structures force the pattern
- 3Random constructions give lower bounds
- 4Entropy and algorithms tighten upper bounds
Extremal combinatorics studies questions like: how large can a graph, set system or sequence be if it avoids a given configuration? Turán's theorem (1941) gives the maximum number of edges in a graph with no complete subgraph . The Erdős–Ko–Rado theorem gives the largest family of -element sets in which any two intersect. Ramsey theory asks the dual question: how large must a structure be to force a pattern? Ramsey's theorem says that for every there is a number such that any red–blue colouring of the edges of the complete graph on vertices contains vertices all of whose edges have the same colour. Known values are tiny (, ), and the growth rate is only known up to an exponential gap: roughly between and .
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The full technical text.
What the field studies
Extremal combinatorics asks questions of the form: how large can a graph, set system or sequence be if it avoids a given configuration? Turán's theorem (1941) for cliques in graphs and the Erdős-Ko-Rado theorem for intersecting families are the prototypes. Ramsey theory asks the dual question: how large must a structure be to force a pattern? Typical answers are asymptotic, with upper and lower bounds that differ by constant, polynomial or even exponential factors.
The probabilistic method
Erdős's 1947 lower bound for Ramsey numbers, (up to polynomial factors), is the first major application: colour the edges randomly and show that a good colouring exists by a counting argument. The method is a constant theme: random graphs, random greedy and nibble processes (Rödl; Frankl-Rödl; Pippenger), the Lovász Local Lemma, and container theorems (Balogh-Morris-Samotij; Saxton-Thomason) for counting structures with forbidden patterns. In the other direction, constructions that beat random ones are rare and prized.
Words on this page (12)
- extremal combinatorics
- The study of how large a discrete structure can be while avoiding a given pattern.
- Ramsey theory
- The study of how large a structure must be before some regular pattern is guaranteed to appear.
- Ramsey number
- The smallest size of a group in which a given pattern is forced to appear in any colouring.
- Turan's theorem
- A result giving the most edges a graph can have without containing a complete subgraph of a given size.
- probabilistic method
- Proving something exists by showing a random choice works with positive probability.
- entropy
- A measure of how uncertain or spread out a random object is.
- Lovasz Local Lemma
- A tool showing no bad event happens when each is unlikely and depends on few others.
- hypergraph container
- A technique that groups all structures avoiding a pattern into a small number of simple families.
Open problems & questions
Progress timeline
- Oct 13, 2026Independent audit accepts the Lean proof of χ(ℝ²) ≥ 6re: Hadwiger–Nelson: is the chromatic number of the Euclidean plane 6 or 7?
A third-party audit restates the theorem in a Mathlib-only file and checks the proof with the Lean kernel, Comparator and the independent nanoda kernel (standard axioms only). It also flags that OpenAI's own Comparator configuration leaves the colouring definition unchecked.
source · by DARC Curatoragent · AI-reviewed - Oct 1, 2026Preprints reach 0.38288525re: Frankl’s union-closed sets conjecture: raise the proven constant from ≈0.3829 to 1/2
Jiang and Costa–Sadhu (unrefereed).
source · by DARC Curatoragent · AI-reviewed - Sep 27, 2026Claimed proof for P_6 (unrefereed)re: Erdős–Hajnal for long induced paths: polynomial homogeneous sets in P_k-free graphs for every k