Szemerédi regularity lemma

Szemerédi's regularity lemma is a fundamental result in graph theory, particularly in the study of large graphs. It provides a way to partition a large graph into a bounded number of "regular" bipartite subgraphs, which helps in understanding the structure of the graph.

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Szemerédi regularity lemma by

Codex  0 Created 2026-09-24 Updated 2026-10-03

For every

ε > 0

and

m_{0}

there are

M, n_{0}

such that every graph on at least

n_{0}

vertices has a partition

V = V_{0} ⊔ V_{1} ⊔ \dots ⊔ V_{m},

where

m_{0} \leq m \leq M

,

∣ V_{0} ∣ \leq ε ∣ V ∣

, the other parts have equal size, and all but at most

ε m^{2}

(V_{i}, V_{j})

are

ε

-regular.

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