Endre Szemerédi

Endre Szemerédi is a Hungarian mathematician known for foundational results in combinatorics and number theory, including Szemerédi’s theorem on arithmetic progressions in dense sets.

Arithmetic progression: A sequence of numbers with a constant difference between consecutive terms. Szemerédi’s theorem concerns long arithmetic progressions inside dense sets of integers.
Szemerédi's theorem: Every subset of the integers with positive upper density contains arbitrarily long arithmetic progressions. This is Szemerédi’s celebrated theorem and the result most closely associated with him.
Paul Erdős: A Hungarian mathematician known for prolific collaboration and major contributions to combinatorics and number theory. Erdős posed influential problems in extremal combinatorics that shaped Szemerédi’s research.
Triangle removal lemma: A theorem relating the number of triangles in a graph to the number of edge deletions needed to make it triangle-free. It is a key consequence of graph regularity and has applications to arithmetic patterns.
Hungarian combinatorics: A tradition of mathematical research in combinatorics associated with Hungarian mathematicians and institutions. Szemerédi’s career emerged from a strong Hungarian combinatorial tradition.
Extremal combinatorics: The study of the largest or smallest possible structures under specified constraints. Many of Szemerédi’s results bound how large a progression-free set can be.
Szemerédi regularity lemma: A graph theorem asserting that every sufficiently large graph admits a bounded partition whose parts are mostly pseudorandomly connected. The lemma bears his name and underpins a wide range of later proofs.
János Komlós: A Hungarian mathematician whose work spans combinatorics, graph theory, and theoretical computer science. Komlós collaborated with Szemerédi on major results in graph theory and algorithms.
Corners theorem: A theorem asserting that dense subsets of a sufficiently large grid contain axis-aligned three-point corners. Ajtai and Szemerédi established this two-dimensional analogue of progression phenomena.
Alfréd Rényi Institute of Mathematics: A research institute of the Hungarian Academy of Sciences based in Budapest. Szemerédi has been a leading researcher at this institute.

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