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A fundamental structural result in extremal graph theory due to Szemerédi (1978). The regularity lemma essentially says that every graph can be well-approximated by the union ...

The blow-up lemma essentially says that regular pairs in Szemerédi's regularity lemma behave like complete bipartite graphs from the point of view of embedding bounded degree ...

Szemerédi's theorem states that every sequence of integers that has positive upper Banach density contains arbitrarily long arithmetic progressions. A corollary states that, ...

A short theorem used in proving a larger theorem. Related concepts are the axiom, porism, postulate, principle, and theorem. The late mathematician P. Erdős has often been ...

Seymour conjectured that a graph G of order n with minimum vertex degree delta(G)>=kn/(k+1) contains the kth graph power of a Hamiltonian cycle, generalizing Pósa's ...

Any lemma on commutative diagrams. It can give relations between maps (as in the five lemma) or tell how to construct new diagrams from old ones (as in the snake lemma).

A diagram lemma which states that, given the above commutative diagram with exact rows, the following holds: 1. If alpha is surjective, and beta and delta are injective, then ...

The three-chord lemma states that in the figure above, l_1+l_2=2lcostheta.

Let p be a non-wandering point of a diffeomorphism S:M->M of a compact manifold. The closing lemma concerns if S can be arbitrarily well approximated with derivatives of ...

A diagram lemma also known as 3×3 lemma. According to its most general statement, the commutative diagram illustrated above with exact rows and columns can be completed by ...