Endre Szemerédi

Endre Szemerédi: In Every Chaos There Is an Order

Six months into medical school, Endre Szemerédi decided he did not trust himself with a scalpel — he doubted his ability, he later said, to "do work bearing such responsibility" — and walked away. He drifted for a while, then enrolled to study mathematics, and was assigned by clerical accident to a doctoral advisor he had never asked for. That misfiled paperwork put him in Moscow under Israel Gelfand, one of the twentieth century's great mathematicians, rather than under Alexander Gelfond, whom Szemerédi had actually wanted. It is one of the field's better accidents.

From Budapest to Moscow by Mistake

Szemerédi was born on August 21, 1940, in Budapest. After his brief and abandoned attempt at medicine, he earned his undergraduate degree at Eötvös Loránd University in Budapest before going on to Moscow State University for his master's and doctoral work. There, under Gelfand — the clerical error that sent him to the wrong supervisor — he began the career in discrete mathematics that would eventually earn him a Fields-Medal-adjacent recognition rarely given for combinatorics: the Abel Prize.

The Theorem That Bears His Name

Szemerédi's defining achievement, announced in 1975, resolved a conjecture that Paul Erdős and Pál Turán had posed decades earlier and that had resisted every attack since: that any set of natural numbers with positive upper density — roughly, any set that does not thin out to nothing relative to the whole number line — must contain arithmetic progressions of every length, no matter how long. It is a statement about hidden order: however irregular or sparse-looking a sufficiently substantial set of integers is, patterned sequences are unavoidably buried inside it. The Abel Prize committee later summarized the spirit of his work in a single phrase: "in every chaos there is an order."

The Regularity Lemma

The proof of that theorem produced something even more consequential than the theorem itself. Along the way, Szemerédi introduced what is now called the Szemerédi regularity lemma, a structural result showing that every sufficiently large graph can be partitioned into a bounded number of pieces that behave, in a precise statistical sense, almost randomly with respect to one another. It sounds like a technical lemma; it became one of the central tools of modern combinatorics, underpinning property testing, graph limit theory, and a large share of extremal graph theory developed in the decades since. Fields Medalists including Timothy Gowers and László Lovász have devoted entire lectures to tracing its consequences — a rare case of a proof's byproduct outgrowing its original theorem in influence.

A Body of Named Results

Szemerédi's career produced an unusually large cluster of results that now carry his name alongside collaborators: the Szemerédi–Trotter theorem in incidence geometry, bounding how many times points and lines in the plane can meet each other; the Hajnal–Szemerédi theorem in graph coloring; the Erdős–Szemerédi theorem on sums and products in finite sets, a foundational result in what became the field of arithmetic combinatorics; and, with Miklós Ajtai, the corners theorem. With Ajtai and János Komlós he established influential bounds on Ramsey numbers and built optimal-depth sorting networks — a rare excursion from pure mathematics into theoretical computer science that helped justify his eventual academic home in a computer science department. With Ajtai, Václav Chvátal, and Monroe Newborn he also proved the crossing number inequality, a foundational bound in graph drawing. Across more than two hundred published papers, the throughline is consistent: finding rigid, unavoidable structure inside objects — sets of integers, graphs, point configurations — that look disordered on their surface.

An American Career Built on Hungarian Foundations

Szemerédi joined Rutgers University in 1986 as State of New Jersey Professor of Computer Science, a position he still holds as emeritus, while remaining permanently affiliated with the Alfréd Rényi Institute of Mathematics in Budapest, the institutional home of Hungarian mathematics since the mid-twentieth century. In between he held visiting posts at Stanford, McGill, the University of South Carolina, and the University of Chicago, as well as a membership at the Institute for Advanced Study in Princeton — a career spanning the Cold War's mathematical migration from Hungary and the Soviet sphere into American research universities.

Recognition

The honors accumulated across five decades: the Alfréd Rényi Prize in 1973 and the George Pólya Prize in 1975, in the years right after his theorem appeared; the Leroy P. Steele Prize and the Rolf Schock Prize in Mathematics, both in 2008; election to the United States National Academy of Sciences in 2010; and, in 2012, the Abel Prize itself, awarded "for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory" — a citation that explicitly credits his work with reshaping fields well beyond combinatorics narrowly defined. Hungary made him a recipient of the Order of Saint Stephen in 2020.

Why Endre Is Called a Genius

The case for Szemerédi's genius is about as clean as mathematics offers: a single proof, of a single long-standing conjecture, generated a tool — the regularity lemma — whose downstream influence on an entire discipline arguably exceeds that of the theorem it was built to prove. That kind of second-order impact, where the method outlives and outgrows its original purpose, is one of the field's clearest markers of a mind operating at the deepest structural level rather than merely solving the problem in front of it. The Abel Prize committee's own language — locating order inside chaos — is not hyperbole; it is a fair paraphrase of what the regularity lemma actually does, and the fact that Gowers and Lovász, themselves among the discipline's most decorated figures, built major lectures around explaining its consequences to other mathematicians is a form of peer acclaim that is hard to manufacture or inflate.

If there is a caveat, it is one of narrowness rather than exaggeration: Szemerédi's fame rests overwhelmingly on results in discrete mathematics and combinatorics, fields that for much of the twentieth century were regarded by parts of the mathematical establishment as less prestigious than analysis, geometry, or number theory in their classical forms — the Abel Prize to a combinatorialist was itself read at the time as an implicit argument that the field deserved recognition it had been slow to receive. That does not diminish the work; it means the "genius" label here tracks a specific, technical form of structural insight rather than the kind of universal renown attached to physicists or mathematicians whose results are famous even outside their field.

Legacy

The regularity lemma is now taught as standard material in graduate combinatorics courses worldwide and remains an active engine of new research, including in areas — theoretical computer science, additive combinatorics, graph limit theory — that barely existed when Szemerédi first wrote it down in 1975. Still active as an emeritus scholar in his mid-eighties, he remains one of the clearest cases in modern mathematics of a single proof's method becoming more consequential than the theorem it was designed to establish.

Achievements

• Abel Prize — 2012
• Affiliated with Rutgers University–New Brunswick
• Educated at Eötvös Loránd University and Lomonosov Moscow State University
• Worked as mathematician, computer scientist and university teacher