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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.


Szemerédi regularity lemma

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For every $\varepsilon>0$ and $m_0$ there are $M,n_0$ such that every graph on at least $n_0$ vertices has a partition
$$
V=V_0\sqcup V_1\sqcup\cdots\sqcup V_m,
$$
where $m_0\leq m\leq M$, $|V_0|\leq\varepsilon|V|$, the other parts have equal size, and all but at most $\varepsilon m^2$ pairs $(V_i,V_j)$ are $\varepsilon$-regular.


Szemerédi regularity lemma

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Szemerédi regularity lemma

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