Description
Szemerédi's Theorem (1975): Every set A ⊆ ℕ with positive upper density contains arbitrarily long arithmetic progressions. That is, if δ̄(A) = lim sup |A ∩ [1,N]|/N > 0, then for every k ≥ 3, A contains a k-term arithmetic progression.
Four proof approaches: (1) Szemerédi's original combinatorial proof (regularity lemma, van der Waerden); (2) Furstenberg's ergodic proof (multiple recurrence); (3) Gowers' Fourier-analytic proof (uniformity norms); (4) Hypergraph regularity (Gowers, Rödl–Schacht). Foundation for the Green–Tao theorem.
Source: Wikipedia; Szemerédi, Furstenberg, Gowers
Dependency Flowchart
Note: Arrows mean "depends on". Multiple proof paths shown.
graph TD
DefDensity["Def: Upper density\nδ̄(A) = lim sup |A∩[1,N]|/N"]
DefAP["Def: k-AP\nArithmetic progression of length k"]
DefPos["Def: Positive density\nδ̄(A) > 0"]
ThmSz["Thm: Szemerédi 1975\nA with δ̄>0 contains k-AP ∀k"]
L1["Lemma: van der Waerden\nMonochromatic k-AP in finite colorings"]
L2["Lemma: Szemerédi regularity\nGraph partition into ε-regular pairs"]
L3["Lemma: Furstenberg multiple recurrence\nErgodic systems, recurrence"]
L4["Lemma: Hypergraph regularity\nGowers, Rödl–Schacht"]
DefUnif["Def: Gowers uniformity U^k\nHigher-order Fourier analysis"]
DefPseudo["Def: Pseudorandom / uniform set\nGowers norm bounds"]
L5["Lemma: Count lemma\nAP count in regular partition"]
DefDensity --> DefPos
DefAP --> ThmSz
DefPos --> ThmSz
L1 --> ThmSz
L2 --> L5
L5 --> ThmSz
L3 --> ThmSz
L4 --> ThmSz
DefUnif --> DefPseudo
DefPseudo --> ThmSz
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
classDef lemma fill:#74c0fc,color:#fff,stroke:#4dabf7
class DefDensity,DefAP,DefPos,DefUnif,DefPseudo definition
class ThmSz theorem
class L1,L2,L3,L4,L5 lemma
Color Scheme
Blue
Definitions
Definitions
Teal
Theorems
Theorems
Purple
Lemmas
Lemmas
Process Statistics
- Nodes: 11
- Edges: 12
- Definitions: 5
- Lemmas: 5
- Theorems: 1